Game Theory & Industrial Organization

• Lebed is the first minor ever prosecuted for stock-market fraud.

• Lebed tools were
  • an America Online (AOL) internet connection,
  • an E*trade account, and
  • four email accounts in Yahoo Finance Message Boards.
• SEC accused him of making his money through a pump and dump strategy.

• Lebed forfeited \($285000\) in profit and interest he had made on \(11\) trades.

• He has never admitted any wrongdoing.
• He kept close to \($500000\) in profit.

People who trade stocks, trade based on what they feel will move and they can trade for profit. Nobody makes investment decisions based on reading financial filings. Whether a company is making millions or losing millions, it has no impact on the price of the stock. Whether it is analysts, brokers, advisors, Internet traders, or the companies, everybody is manipulating the market. If it wasn't for everybody manipulating the market, there wouldn't be a stock market at all…

(Jonathan Lebed, statement to his lawyer, Lewis, 2001)

• The monopolist sets the price of the commodity it produces using a markup.
• The monopolist pricing rule can be calculated by \[p_{m} = \frac{\text{Marginal Cost}}{\frac{1}{\text{Demand Elasticity}} + 1}\]

1. Consider a market with inverse demand given by \(p(q) = p_{0} + p_{1}q\) with \(p_{0}>0\) and \(p_{1}<0\).

   1. Calculate the demand function (i.e., invert \(p\)). For which values of price is demand non-negative?
   2. Calculate the demand elasticity as a function of price using the demand function. What is the sign of elasticity?
   3. Calculate the demand elasticity as a function of quantity using the inverse demand function.
   4. Substitute the quantity variable in the result of part (c) with the demand calculated in part (a) and verify that you get the result of part (b).
   5. Calculate the quantity for which demand exhibits unitary elasticity.
   1. Fix a value for \(p\) and solve for \(q\) to get \[q(p)=-\frac{p_{0}}{p_{1}} + \frac{1}{p_{1}} p.\] Since \(p_{1}<0\), demand is non-negative if and only if \(p \le p_{0}\).

   2. We calculate

      \begin{align*} E(p) &= \frac{\mathrm{d} q}{\mathrm{d} p} \frac{p}{q(p)} \\ &= q'(p) \frac{p}{q(p)} \\ &= \frac{1}{p_{1}} \frac{p}{-\frac{p_{0}}{p_{1}} + \frac{1}{p_{1}} p} \\ &= \frac{p}{-p_{0} + p} \end{align*}

      For every \(0 \le p\le p_{0}\) is non-positive.

   3. We have

      \begin{align*} E(q) &= \frac{\mathrm{d} q}{\mathrm{d} p} \frac{p(q)}{q} \\ &= \frac{1}{\frac{\mathrm{d} p}{\mathrm{d} q}} \frac{p(q)}{q} \\ &= \frac{1}{p'(q)} \frac{p(q)}{q} \\ &= \frac{1}{p_{1}} \frac{p_{0} + p_{1}q}{q} \end{align*}

   4. Substituting gives

      \begin{align*} E(q(p)) &= \frac{p_{0} + p_{1}q(p)}{p_{1} q(p)} \\ &= \frac{p_{0} - p_{0} + p}{- p_{0} + p} \\ &= \frac{p}{-p_{0} + p} \end{align*}
   5. Using part (c), we have \(E(q)=-1\) if and only if \[p_{0} + p_{1}q = -p_{1}q,\] or equivalently \[q = -\frac{p_{0}}{2p_{1}}.\]

2. Consider a monopolist in a market with inverse demand \(p(q) = p_{0} + p_{1}q\), where \(p_{0}>0\) and \(p_{1} < 0\). The monopolist's cost function is given by \(c(q) = c_{1}q\), where \(0 < c_{1}\).

   1. Set up the maximization problem of the monopolist for which the monopolist chooses the produced quantity.
   2. Calculate the monopoly quantity, price, and profit.
   3. Instead, suppose that the monopolist is choosing prices. Calculate the demand function and setup the maximization problem.
   4. Calculate the quantity, price, and profit. Compare with the maximization problem over quantities. Are the results the same? Why, or why not?
   5. Calculate the consumer welfare and the total welfare.
   6. Compare with the welfare of the perfect competition case and calculate the deadweight loss.
   1. The monopolist solves \[\max_{q} \left\{ p(q)q - c(q) \right\}\]

   2. The first order condition is \[p'(q)q + p(q) = c'(q) \iff p_{0} + 2p_{1}q = c_{1},\] which gives

      \begin{align*} q_{m} &= -\frac{p_{0} - c_{1}}{2p_{1}}, \\ p_{m} &= \frac{c_{1} + p_{0}}{2}, \end{align*}

      and \[\pi_{m} = -\frac{(p_{0} - c_{1})^{2}}{4p_{1}}.\]

   3. The demand function is \[q(p) = -\frac{1}{p_{1}} (p_{0} - p)\] and the monopolist problem is \[\max_{p} \left\{ q(p)p - c(q(p)) \right\}.\]
   4. The first order condition is \[q'(p)p + q(p) = c'(q)q'(p) \iff \frac{1}{p_{1}}(2p - p_{0}) = \frac{c_{1}}{p_{1}},\] which leads to the same price, quantity, and profit as in part (b). The result is the same because inverse demand is affine (more generally because inverse demand is one-to-one).

   5. The consumer welfare is

      \begin{align*} C_{m} &= \int_{0}^{q_{m}} (p(q) - p_{m}) \mathrm{d} q \\ &= p_{0}q_{m} + \frac{p_{1}}{2} q_{m}^2 - p_{m} q_{m} \\ &= \frac{1}{2} q_{m} (p_{0} - p_{m})\\ &= -\frac{(p_{0} - c_{1})^{2}}{8p_{1}}, \end{align*}

      so the total welfare is \[W_{m} = C_{m} + \pi_{m} = - 3 \frac{(p_{0} - c_{1})^{2}}{8p_{1}}.\]

   6. In perfect competition, \(p_{c} = c_{1}\). Then, \[q_{c}=\frac{c_{1} - p_{0}}{p_{1}}\] and profits are zero. The competition total welfare is

      \begin{align*} W_{c} &= C_{c} \\ &= \int_{0}^{q_{c}} (p(q) - c_{1}) \mathrm{d} q \\ &= p_{0}q_{c} + \frac{p_{1}}{2} q_{c}^2 - c_{1} q_{c} \\ &= \frac{1}{2} q_{c} (p_{0} - p_{c})\\ &= -\frac{(p_{0} - c_{1})^{2}}{2p_{1}}. \end{align*}

      Finally, the deadweight loss is \[D = W_{c} - W_{m} = -\frac{(p_{0} - c_{1})^{2}}{8p_{1}}.\]

• In the games that we will examine, the utilities and the choices of players are common knowledge.
• Common knowledge is information that is known and understood in the same way by all the players of a game.
• There is an element of infinite recursion in the idea of common knowledge.

• The agents of a game have common knowledge of a property \(P\) when they all know \(P\), they all know that they know \(P\), they all know that they all know that they know \(P\), etc.

• Each player in a game has one or more decisions to make.

• A single choice made at a particular decision node is called an action.
• The collection of all actions of a player in a game is called a pure strategy (or simply strategy when it is understood from context that it is pure).
• In games where a player has a single decision to make, her actions and strategies coincide.

• Simple games can be represented using a table documenting the primitives of the game.
• This representation is called the normal form of a game.

• The extensive form of a game is a representation in terms of a tree.

• The extensive form depicts more information than the normal form:

  • It illustrates the order in which the players act.
  • It illustrates the information available to each player.

• If a player has the same information at two (or more) nodes, the nodes are connected with a dotted line.
• An information set is a collection of decision nodes that the player making decisions cannot distinguish at the time of decision.
• The player knows that she is located at one of the nodes of the information set, but she does not know at which one of them.

• On some occasions, a player can choose a strategy that makes her better off irrespective of the strategies chosen by other players.
• A strategy that, for all strategies other players can choose, gives a higher payoff to a player compared to every other strategy available to her is called a dominant strategy.

• \(Left \succ_{B} Right\)

• Nash equilibria do not say how, why, or whether these strategies are reached in a game.
• The definition of Nash equilibria suggests that if they are reached, then there is no incentive for anyone to change her behavior.

• Nash equilibria are not unique.
• Multiple situations in a game may constitute points from which no one wants to deviate.
• The payoffs of the players in different Nash equilibria can be significantly different.

• \(NE = \left\{ \left\{ Bottom, Left \right\}, \left\{ Top, Right \right\} \right\}\)

• No. Nash equilibria can be Pareto inefficient.
• A classic example is the prisoner's dilemma.

• The grappler has two pure strategies (\(Command\ Grab\) and \(Normal\ Grab\)).
• The mixed strategy \((p, 1-p)\) assigns probability \(p\) to choosing \(Command\ Grab\) and probability \(1-p\) to choosing \(Normal\ Grab\).

• Finite games always have at least one Nash equilibrium in mixed strategies (Nash, 1950).
• A finite game is a game in which the number of players, actions, and decision nodes are finite.

1. Consider the Bach or Stravinsky game (also known as Battle of the Sexes, which is outdated in current norms). A couple wishes to go to a music concert by either Bach or Stravinksy. One partner prefers Bach and the other Stravinsky. Nevertheless, their primary concern is to go out together.

   1. Find all the Nash equilibria.
   2. How do you interpret the objective of the players in this game?

   1. There are two pure Nash equilibria in this game, namely \((Bach, Bach)\) and \((Stravinsky, Stravinsky)\).

      There is also a mixed strategy equilibrium in which player \(A\) chooses \(Bach\) with \(p=\frac{2}{3}\) and \(Stravinksy\) with \(p=\frac{1}{3}\), while player \(B\) chooses \(Bach\) with \(q=\frac{1}{3}\) and \(Stravinksy\) with \(q=\frac{2}{3}\).

   2. The objective of the players is to coordinate. The players would like to choose the same action. Games of this type are called coordination games.

1. Consider the Hawk and Dove game. Two animals fight over a food source with a value of \(1\). Each animal can behave aggressively (like a Hawk) or peacefully (like a Dove). If both animals behave peacefully, they split the food. If only one animal behaves aggressively, it gets all the food. If both animals act aggressively, the value of the food source is reduced by \(c>0\) (lost in the fight), and the animals split the remainder equally.

   1. Write the normal form of the game.
   2. Find all the Nash equilibria for every value of the cost parameter \(c\).
   3. How do you interpret the equilibria of each case?
   1. The normal for of the game is as follows.

   2. We distinguish three cases based on the value of the parameter \(c\). For \(01\), the game has two pure strategy Nash equilibria. These are \(\{Dove, Hawk\}\) and \(\{Hawk, Dove\}\). The last case combines the pure strategy equilibria of the first and second cases. That is, for \(c=1\), the equilibria in pure strategies are \(\{Hawk, Hawk\}\), \(\{Dove, Hawk\}\), and \(\{Hawk, Dove\}\)

      To calculate the mixed equilibria, suppose that Animal \(A\) mixes its actions by choosing \(Dove\) with probability \(p\). In equilibrium, Animal \(B\) should be indifferent between choosing \(Dove\) and \(Hawk\), therefore, \[p\frac{1}{2} = p + (1-p)\frac{1}{2}(1 -c).\] Solving for \(p\) gives \[p = \frac{c-1}{c}.\] Thus, if \(c < 1\), there is no mixed strategy equilibrium. If \(c=1\), then \(p=0\), so we essentially get one of the pure strategy equilibria of this case (namely \(\{Hawk, Hawk\}\)). Lastly, if \(c>1\), we get an additional mixed equilibrium.

   3. If \(c<1\), the cost of fighting is small, and the animals attempt to get the value for their own. This leads to a prisoners' dilemma type of game with Pareto inefficient outcome. If \(c>1\), the cost of fighting is great enough to deter fighting. The animals then resort to playing a coordination game (Similar to Bach or Stravinksy).

• In the early 1980s, the common practice of operating system companies was to charge hardware manufacturers for each operating system copy installed in a computer.

• Microsoft offered an alternative plan.
  • Charge computer manufacturers based on (the past number of) built computers.
  • The manufacturer was paying a general licensing fee and then could install the operating system in all the computers it produced.
  • Microsoft was offering low-priced licensing contracts making their operating system (MS-DOS) very attractive to manufacturers.

• Effectively, manufacturers could purchase Microsoft's operating system at much lower prices than the operating systems of other software companies.
  • A manufacturer had to pay \($50\ -\ $100 \) for installing an alternative operating system on an additional machine.
  • It cost nothing to install MS-DOS on an additional machine once a licensing contract with Microsoft has been signed.
  • MS-DOS ended up being the default operating system

• Android is a community (open source) operating system for mobile devices based on the Linux kernel.
• The wide use of smartphones and tablets drastically changed the operating system market.
• Although Microsoft offered an operating system suitable for smartphones and tablets, it did not manage to keep its primacy.

• Android is free. Anyone can install the operating system on her device after accepting the terms and conditions.
• For mobile device manufacturers, Android is a cheaper operating system alternative for their products.
• This leads to more competitive prices for consumers too.
  • Microsoft's operating systems lost their primacy in the overall operating system market in \(2017\).
  • Microsoft's operating systems are still dominant in less portable devices, such as desktop PCs and Laptops.
  • The rise of Android has also impacted Microsoft's pricing strategies for its desktop operating systems.
  • Licensed users were able to upgrade to the last two versions of Microsoft's operating system without paying for a new license.

• Oligopolies may compete using pricing strategies or by choosing quantities.
• Different means of competition strategies crucially affect the market outcome.
• The means of competition is a decisive component of the market structure.

• Oligopolies have used explicitly written collusive strategies in the past (e.g., cartels).
• Nowadays, collusion is usually illegal.
• Instances of tacit collusion have also been documented.
• Tacit collusion strategies do not require explicit contracts or communication.

• For inverse demand and costs given by

\begin{align*} p(q) &= p_{0} + p_{1} q \quad &&(p_{0}>0,\ p_{1}<0)\\ c(q) &= c_{1} q \quad &&(c_{1}>0). \end{align*}

• The best responses become

\[q_{i} = \frac{c_{1} - p_{0} - p_{1} q_{j}}{2 p_{1}}.\]

• The equilibrium quantities are given by

\[q_{i} = \frac{c_{1} - p_{0}}{3 p_{1}}.\]

• The profits are symmetric and can be calculated as

\[\pi_{i} = -\frac{(c_{1} - p_{0})^{2}}{9 p_{1}}.\]

• Suppose inverse demand is given by

\begin{align*} p(q) &= p_{0} + p_{1} q . \end{align*}

• All firms have (symmetric) costs

\begin{align*} c(q) &= c_{1} q . \end{align*}

• The best responses become

\[q_{i} = \frac{c_{1} - p_{0} - p_{1} \sum_{j\neq i} q_{j}}{2 p_{1}}.\]

• The equilibrium quantities are given by

\[q_{i} = \frac{c_{1} - p_{0}}{(n + 1) p_{1}}.\]

• The total quantity is

\[q = \sum_{i=1}^{n} q_{i} = \sum_{i=1}^{n} \frac{c_{1} - p_{0}}{(n + 1) p_{1}} = \frac{n}{n+1}\frac{c_{1} - p_{0}}{p_{1}},\]

• So, the market price becomes

\[p(q) = p_{0} + p_{1} n \frac{c_{1} - p_{0}}{(n + 1) p_{1}} = \frac{n c_{1} + p_{0}}{n + 1}.\]

• Profits are then

\[\pi_{i} = -\frac{(c_{1} - p_{0})^{2}}{(n + 1)^{2} p_{1}}.\]

• The case of two firms can be obtained by replacing \(n=2\) in the above results.

• We can also obtain the solution of the monopoly problem if we set \(n=1\).
• With \(n=1\), we get the monopolistic profit

\[\pi_{1} = -\frac{(c_{1} - p_{0})^{2}}{4 p_{1}}.\]

• Profits decrease as the number of firms in the market increases.
• The limiting case is

\[\pi_{i} \xrightarrow[n\to \infty]{} 0.\]

• In addition, \(p(q) \to c_{1}\), \(q_{i}\to 0\), and \(q\to\frac{c_{1}-p_{0}}{p_{1}}\) as \(n\to\infty\).
• The production of each firm becomes negligible, and the total market quantity and price approach those of perfect competition.

• The necessary condition for each firm is

\[28 - 2 q_{j} - 4 q_{i} = 4.\]

• Solving for \(q_{i}\) gives the best response of firm \(i\)

\[q_{i} = \frac{24 - 2 q_{j}}{4}.\]

• Combining the two best responses gives the Nash equilibrium

\[q_{i} = 4 = q_{j}.\]

• Profits are then

\[\pi_{i} = 32.\]

• Suppose that firm \(j\) sets a price \(p_{j}\) that is greater than the marginal cost of firm \(i\) (i.e., \(4\)).
• Firm \(i\) can undercut by a small amount and grab all the market. For instance, set price \(p_{i} = \frac{p_{j} + 4}{2}\).
• Thus, firm \(j\) can only set a price equal to firm \(i\)'s marginal cost.
• Analogous arguments hold for firm \(i\)'s pricing strategy.

• The only possible equilibrium is to set prices equal to the (common) marginal cost.
• Firms do not have any incentive to deviate.
• Setting lower prices leads to losses.
• Setting higher prices leads to zero profits.
• Even with two firms, price competition leads to prices similar to perfect competition.

• If firm \(2\) chooses \(x_{2} > \frac{1}{2}\), firm \(1\) would like to undercut by a small amount and set \(x_{1} = x_{2} - \varepsilon > \frac{1}{2}\).
• Then firm \(2\) has a profitable deviation by changing to \(x_{2} = \frac{1}{2}\).
• Thus any \(x_{2} > \frac{1}{2}\) cannot be an equilibrium.
• Similarly, any \(x_{2} < \frac{1}{2}\) cannot be an equilibrium.
• Analogous arguments hold for firm \(1\) because of symmetry.

• Therefore, the only possible equilibrium is \(x_{1} = \frac{1}{2} = x_{2}\).
• Firms split the market and make equal profits.
• Any deviation leads to fewer profits for the firm that moved.

1. Consider a Cournot duopoly with inverse demand \(p(q) = p_{0} + p_{1} q\) and costs \(c_{i}(q) = c_{1, i} q\), for \(i = \{1, 2\}\), where \(p_{0},c_{1}>0\) and \(p_{1}<0\). What happens if costs are not symmetric? Which firm produces more? Which firm makes more profit?

   The best responses become \[q_{i} = \frac{c_{1, i} - p_{0} - p_{1} q_{j}}{2 p_{1}}.\] Equilibrium ceases to be symmetric. The equilibrium quantities are given by \[q_{i} = \frac{2c_{1, i} - c_{1, j} - p_{0}}{3 p_{1}}.\] The symmetric equilibrium quantities are obtained as a special case from the last formula by setting \(c_{1, i} = c_{2, j}\).

   Firm \(i\) produces more than \(j\) if and only \[\frac{2c_{1, i} - c_{1, j} - p_{0}}{3 p_{1}} \ge \frac{2c_{1, j} - c_{1, i} - p_{0}}{3 p_{1}},\] which, because \(p_{1} < 0\), is equivalent to \(c_{1, i} \le c_{1, j}\). Thus, the lower cost firm produces more.

   The total market quantity is \[q = \frac{2c_{1, i} - c_{1, j} - p_{0}}{3 p_{1}} + \frac{2c_{1, j} - c_{1, i} - p_{0}}{3 p_{1}} = \frac{c_{1, i} + c_{1, j} - 2p_{0}}{3 p_{1}},\] and the market price \[p(q) = \frac{c_{1, i} + c_{1, j} + p_{0}}{3}.\] We can the calculate the profit of firm \(i\) as

   \begin{align*} \pi_{i} &= \left(\frac{c_{1, i} + c_{1, j} + p_{0}}{3} - c_{1, i}\right) \frac{2c_{1, i} - c_{1, j} - p_{0}}{3 p_{1}} \\ & = -\frac{(2c_{1, i} - c_{1, j} - p_{0})^{2}}{9 p_{1}}. \end{align*}

   The firm that produces more makes the greatest profit. The easiest way to get this result is to rewrite profits as \(\pi_{i} = -q_{i}^{2}p_{1}\). Since \(p_{1}<0\), we have \(\pi_{i}\ge \pi_{j}\) if and only if \(q_{i}\ge q_{j}\).

1. Consider a differentiated product duopoly with firms competing in prices. The market consists of a homogeneous unit mass of consumers. Each consumer \(h\in[0,1]\) has unit demand for commodity \(i=1,2\). The (gross) valuation of each consumer for commodity \(i\) is \(v_{i}\) with \(v_{1} > v_{2}\). If the net (gross minus price) valuations of the two commodities are equal, commodity \(1\) gets all the demand. The marginal production costs of both firms are zero.

   1. Suppose that each consumer can only buy one of the two products. Find the Nash equilibria.
   2. Suppose that consumers can also buy both products in a bundle. The bundle has valuation \(v_{b}\) with \(v_{1} + v_{2} > v_{b} > v_{1}\). Find the Nash equilibrium.
   3. Calculate the consumer surpluses of the two cases and compare them.

   1. The Nash equilibrium is characterized by \(p_{1}^{\ast} = v_{1}-v_{2}>0\) and \(p_{2}^{\ast}=0\). We work our way to this result by showing that there is no profitable unilateral deviation. In the candidate equilibrium, the profits are \(\pi_{1}^{\ast} = v_{1}-v_{2}\) and \(\pi_{2}^{\ast} = 0\).

      Suppose that firm \(2\) deviates to \(p_{2}>0\). We then have \(p_{1} < p_{2} + v_{1} - v_{2}\). This implies that \(v_{2} - p_{2} < v_{1} - p_{1}\), so the consumers prefer commodity \(1\). As a result, \(\pi_{2} = 0 = \pi_{2}^{\ast}\). Thus, firm \(2\) has no profitable deviation.

      Consider the deviations of firm \(1\). There are two potential deviations. Firstly, suppose that firm deviates to \(p_{1} > v_{1} - v_{2} = p_{2} + v_{1} - v_{2}\). Then, commodity \(2\) is preferred, and firm \(1\) makes zero profit, which is less than \(\pi_{1}^{\ast}\). Secondly, consider a deviation \(p_{1} < v_{1} - v_{2} = p_{2} + v_{1} - v_{2}\). Then, commodity \(1\) is preferred, and firm \(1\) earns \(\pi_{1} = p_{1} < v_{1} - v_{2} = \pi_{1}^{\ast}\).

   2. In equilibrium, it holds \[v_{1}-p_{1}^{\ast\ast} = v_{2}-p_{2}^{\ast\ast} = v_{b}-p_{1}^{\ast\ast}-p_{2}^{\ast\ast}\] because at least one of the two firms has a profitable deviation otherwise. For example, suppose that \(v_{1}-p_{1} < v_{2}-p_{2}^{\ast\ast}\). Then, only commodity \(2\) is consumed, and the profit of firm \(1\) is zero. Firm \(1\) can deviate to \(p_{1}^{\ast\ast}\) and earn profit \(p_{1}^{\ast\ast} = v_{b} - v_{2} > 0\). Similarly, other cases can be excluded.
   3. The consumer surplus of the market in part one is \[C^{\ast} = v_{1} - p_{1}^{\ast} = v_{1} - v_{1} + v_{2} = v_{2}.\] The consumer surplus of the market in part two is \[C^{\ast\ast} = v_{1} - p_{1}^{\ast\ast} = \underbrace{v_{1} - v_{b}}_{<0} + v_{2} > 0.\] Therefore, \(C^{\ast}>C^{\ast\ast}\) and consumers are better off in the first case.

• Commodus was the Roman emperor from 176 to 192 AD.

• He was assassinated in 192 AD by a wrestler in a bath.
• The assassination conspiracy was directed by one of his most trusted Praetorians, Quintus Aemilius Laetus.
• Pertinax, who was also a conspirator, ascended to power and paid the Praetorian guards \(3000\) denarii premium.

• Pertinax was assassinated three months later by a group of guards!

• The Praetorian guards put the empire up to auction, taking place at the guards' camp.
• The throne was to be given to the one who paid the highest price.
• Titus Flavius Claudius Sulpicianus, Pertinax's father-in-law, made offers for the throne (in the camp).
• Didius Julianus arrived at the camp finding the entrance barred.
• He started shouting out offers to the guards.

• Julianus ruled for \(66\) days!

• He was very unpopular with the Roman public because he acquired the throne by paying instead of conquering it.
• He was killed in the palace… by a soldier.
• His last words were: "But what evil have I done? Whom have I killed?"

• Consider the static prisoner’s dilemma again.
• There is a unique Nash equilibrium in which players do not coordinate (i.e., they both confess).

• Can this change if the game is repeated?

• Suppose that the static prisoner's dilemma is used as the stage game of a repeated game with an infinite time horizon.
• The payoff of all future dates is discounted by \(0 <\delta < 1\).

• One equilibrium is to play the stage game's Nash equilibrium at each date.
• Player \(i\) gets a payoff of \(-3\) at each date, so

\[u_{n,i} = \sum_{t=0}^{\infty} \delta^{t}(-3)=-\frac{3}{1-\delta}\]

• A trigger strategy is a strategy, where at each iteration of a repeated game an action is selected based on the coordination state of the game. If all players coordinated in the past then a coordinating action is chosen for the current iteration. Instead, past defections trigger players to choose punishing actions for the current iteration.
• Trigger strategies can be further specified as
  • Grim trigger strategies, where the punishment continues indefinitely after a player defects.
  • Tit for tat strategies, where the punishment is only applied for a limited number of dates after a defection.

• Players can coordinate by using trigger strategies.
• If at all previous dates the other player has denied, then deny. Otherwise, confess.
• If players coordinate, then each gets a payoff of \(-1\) at each date, thus

\[u_{c,i} = -\frac{1}{1 - \delta}\]

• If player \(i\) deviates at the current date, then her payoff at the current date is \(0\).
• At every subsequent date, her payoff is \(-3\), because her past deviation triggers the other player's punishment.
• Therefore,

\[u_{d,i} = -3\frac{\delta}{1 - \delta}\]

• Coordination can be supported if such a deviation is not profitable, i.e.

\[u_{c,i} \ge u_{d,i} \iff \delta \ge \frac{1}{3}\]

• As long as players are patient enough, the underlying threats of trigger strategies make coordination feasible.

• \(SPE = \left\{ \left\{Bottom, \left(Left', Right \right)\right\}, \left\{Bottom, \left(Right', Right \right)\right\}\right\}\)

• A subgame of a dynamic game is the restriction of the game starting from a particular decision node and including all subsequent decision nodes and branches (actions) of the original game.
• A collection of strategies, one for each player, such that its restriction to each subgame of the original game constitutes a Nash equilibrium of this subgame is called subgame perfect equilibrium.
• This implies that the past does not matter in optimal decisions once a decision node is reached.
• Equilibria of this type are typically abbreviated as SPE.
• Backward induction can be used to calculate subgame perfect equilibria.

• There are two players negotiating how to share a surplus of unit value.
• Player \(A\) moves first and makes an offer \(x\in[0,1]\).
• Player \(B\) moves second and decides whether to accept or reject the offer.
• If the offer is accepted, player \(A\) gets \(x\), and player \(B\) gets \(1-x\).
• If the offer is rejected, both players get zero.

• Player \(B\) accepts if \(1-x \ge 0\) or, equivalently, if \(x\le 1\).
• Player \(A\) offers \(x = 1\).

• Suppose that player \(B\) can make a counteroffer.
• She can offer \(y\in[0,1]\) if she rejects the offer of player \(A\).
• If the counteroffer is accepted, player \(A\) gets \(1-y\), and player \(B\) gets \(y\).
• If the counteroffer is rejected, both players get zero.

• Player \(A\) accepts the counteroffer if \(1-y \ge 0\) or, equivalently, if \(y\le 1\).
• Player \(B\) offers \(y = 1\).
• Player \(B\) accepts the first offer if \(1-x \ge y = 1\) or , equivalently, if \(x \le 0\).
• Player \(A\) offers \(x = 0\).

• Suppose that the counter-proposal game is infinitely repeated until a deal is reached.
• Every time a player rejects an offer, she makes a counteroffer.
• Players discount every offer-round with factors \(\delta_{A},\delta_{B}\in[0,1)\).

• Suppose the game is at date \(t\) and player \(B\) makes an offer.
• Player \(A\) accepts if \(y\le 1\), so player \(B\) offers \(y=1\).
• At date \(t-1\), player \(B\) accepts if \(x\le 1-\delta_{B}\), so player \(A\) offers \(x=1-\delta_{B}\).
• At date \(t-2\), player \(A\) accepts if \(y\le 1 - \delta_{A} + \delta_{A}\delta_{B}\), so player \(B\) offers \(y=1 - \delta_{A} + \delta_{A}\delta_{B}\).
• Recursively, one can show that player \(A\) offers

\[x = \frac{1-\delta_{B}}{1-\delta_{A}\delta_{B}}.\]

• If player \(A\) becomes more patient (\(\delta_{A}\ \uparrow\))
  • she is more willing to postpone acceptance for the next date
  • player \(B\) loses bargaining power
  • player \(A\) gets a greater part of the surplus (\(x\ \uparrow\))
• If player \(B\) becomes more patient (\(\delta_{B}\ \uparrow\))
  • she is more willing to postpone acceptance for the next date
  • player \(A\) loses bargaining power
  • player \(A\) gets a smaller part of the surplus (\(x\ \downarrow\))

1. Consider a two-player game in which, at date \(1\), player \(A\) selects a number \(x>0\), and player \(B\) observes it. At date \(2\), simultaneously and independently, player \(A\) selects a number \(y_{A}\) and player \(B\) selects a number \(y_{B}\). The payoff functions of the two players are \(u_{A} = y_{A} y_{B} + x y_{A} - y_{A}^{2} - \frac{x^{3}}{3}\) and \(u_{B} = -( y_{A} - y_{B} )^{2}\). Find all the subgame perfect Nash equilibria.

   We find the subgame perfect equilibrium using backward induction. At the last date, player \(B\) solves \[\max_{y_{B}} \left\{-(y_{A}-y_{B})^{2}\right\}\] and gets the best response \(y_{B}(y_{A})=y_{A}\). At the same date, player \(A\) solves \[\max_{y_{A}} \left\{y_{A}y_{B} + x y_{A} - y_{A}^{2} - \frac{x^{3}}{3}\right\}\] to get the best response \(y_{A}(y_{B};x)=x\). Solving the system of best responses, we find that \(y_{A}^{\ast}=y_{B}^{\ast}=x\) in equilibrium.

   At the initial date, player \(A\) anticipates the behavior of date \(2\) and maximizes \[\max_{x} \left\{ x^{2} - \frac{x^{3}}{3}\right\}.\] This gives the first order condition \(2x - x^{2}=0\), from which we get the solution \(x = 2\).

   Consequently, we have a unique Nash equilibrium given by \(\left\{(x=2,y_{A}=2), y_{B}=2\right\}\).

2. Find the subgame perfect equilibria of the following game. How many subgames does the game have?

   There are two subgames. The subgame perfect equilibrium is \(\left\{(U,E,H),(B,C)\right\}\).

1. Consider the alternating offer bargaining game with two players \(A\) and \(B\) negotiating how to split surplus with a unitary value. Player \(A\) moves first and makes an offer \(x\in[0,1]\). Player \(B\) moves second and decides whether to accept or reject the offer. If the offer is accepted, player \(A\) gets \(x\), and player \(B\) gets \(1-x\). If the offer is rejected, player \(B\) makes a counteroffer \(y\in[0,1]\). If the counteroffer is accepted, player \(A\) gets \(1-y\), and player \(B\) gets \(y\). If the counteroffer is rejected, the game continues with player \(A\) making an offer. Players discount every offer-round with factors \(\delta_{A},\delta_{B}\in[0,1)\).

   1. Calculate the recursive subgame perfect equilibrium.
   2. Show that if the discount factor of player \(A\) increases, then she can get a greater part of the surplus.
   3. Show that if the discount factor of player \(B\) increases, then player \(A\) gets a smaller part of the surplus.

   1. Suppose that we are at time \(t\) and player \(B\) makes an offer. Player \(A\) accepts the offer if and only if \(1-y \ge 0\). Thus, player \(B\), aiming at maximizing her share, solves \[\max_{1-y \ge 0} y\] and offers \(y=1\). At time \(t-1\), player \(B\) knows that if the game proceeds to time \(t\), player \(A\) will accept the offer \(y=1\). Therefore, if she waits for the next time point, player \(B\) can gain \(\delta_{B}\cdot 1\) in present value terms. Hence, she accepts any offer of player \(A\) securing her at least this gain, namely \(1-x \ge \delta_{B}\). Player \(A\) anticipates this behavior, solves \[\max_{1-x \ge \delta_{B}} x,\] and offers \(x=1-\delta_{B}\).

      At time \(t-2\), player \(A\) knows that if she delays, she can gain \(1-\delta_{B}\), which corresponds to gaining \(\delta_{A}(1-\delta_{B})\) discounted in time \(t-2\) terms. She, therefore, accepts an offer if \(1-y \ge \delta_{A} - \delta_{A}\delta_{B}\). Thus, player \(B\) solves \[\max_{1-y \ge \delta_{A} - \delta_{A}\delta_{B}} y\] and offers \(y = 1 - \delta_{A} + \delta_{A}\delta_{B}\). In a similar manner, one argues that, at time \(t-3\), player \(A\) solves \[\max_{1-x \ge \delta_{B} - \delta_{A}\delta_{B} + \delta_{A}\delta_{B}^2} x,\] and offers \(x=1 - \delta_{B} + \delta_{A}\delta_{B} - \delta_{A}\delta_{B}^2\).

      Repeating the aforementioned recursive arguments, we find that player \(A\)'s offer at date \(t-(2n-1)\) (observe that, because we started with player \(B\), player \(A\) moves at odd offer-rounds) is

      \begin{align*} x &= 1 - \delta_{B} + \delta_{A}\delta_{B} - \delta_{A}\delta_{B}^2 + \dots + \delta_{A}^{n-1}\delta_{B}^{n-1} - \delta_{A}^{n-1}\delta_{B}^{n} \\ &= 1 + \dots + \delta_{A}^{n-1}\delta_{B}^{n-1} - \delta_{B}\left(1 + \dots + \delta_{A}^{n-1}\delta_{B}^{n-1}\right) \\ &= \left(1 - \delta_{B}\right) \sum_{j=0}^{n-1} \delta_{A}^{j}\delta_{B}^{j}. \end{align*}

      Letting \(n\) go to infinity, the share of player \(A\) converges to

      \begin{align*} x = \frac{1 - \delta_{B}}{1 - \delta_{A}\delta_{B}}. \end{align*}

   2. We can find the effect of \(\delta_{A}\) on player \(A\)'s share by calculating

      \begin{align*} \frac{\partial x}{\partial \delta_{A}} = \delta_{B}\frac{1 - \delta_{B}}{\left(1 - \delta_{A}\delta_{B}\right)^{2}}, \end{align*}

      which is positive.

   3. From

      \begin{align*} \frac{\partial x}{\partial \delta_{B}} = - \frac{1 - \delta_{A}}{\left(1 - \delta_{A}\delta_{B}\right)^{2}} < 0, \end{align*}

      we see that player \(A\)'s share is decreasing in \(\delta_{B}\).

• Ford and General Motors are two of the largest automobile companies worldwide.
• They were both founded in the beginning of the \(20^{\text{th}}\) century.
• Since then, they have been competing for the lead in the US automobile market.

• Model T's production innovations made Ford the leading car company from about 1910 to 1925.

• The reluctance of Ford to keep introducing innovations and the absence of flexibility changed the situation.
• In 1925, GM surpassed Ford in total revenue and kept ahead until 1986.
• Eventually in 1927, Ford shutdown for 6 months to update its production lines for Model A.
• GM also surpassed Ford in total sales during this period.

• The leader chooses the quantity it produces first.
• Thus, the follower does not face the entire market demand but only the demand remaining after the leader's choice.
• The difference between the market demand and the aggregated supplied quantities of all competitive firms is a firm's residual demand.

• We solve the problem with backward induction.
• The follower claims the residual demand.
• For any given quantity \(q_{1}\) produced by the leader, the follower's problem is

\[\max_{q_{2}} \left\{ p(q_{1} + q_{2})q_{2} - c_{2}(q_{2}) \right\}.\]

• Interior optimal solutions require

\[p'(q_{1} + q_{2})q_{2} + p(q_{1} + q_{2}) = c_{2}'(q_{2}).\]

• From this condition, we obtain (solving for \(q_{2}\)) the best response of the follower, say

\[q_{2} = b_{2}(q_{1}).\]

• The leader anticipates the best response of the follower and solves

\[\max_{q_{1}} \left\{ p(q_{1} + b_{2}(q_{1}))q_{1} - c_{1}(q_{1}) \right\}.\]

• Its optimality condition is given by

\[p'(q_{1} + b_{2}(q_{1})) \left(1 + b_{2}'(q_{1})\right) q_{1} + p(q_{1} + b_{2}(q_{1})) = c_{1}'(q_{1}).\]

• Suppose that the demand and cost functions are given by

  \begin{align*} p(q) &= p_{0} + p_{1} q, \\ c_{1}(q_{1}) &= c q_{1}, \\ c_{2}(q_{2}) &= c q_{2}. \end{align*}

• The follower's necessary condition becomes

\[p_{1}q_{2} + p_{0} + p_{1}(q_{1} + q_{2}) = c.\]

• Then, the best response of the follower is

\[b_{2}(q_{1}) = \frac{c - p_{0} - p_{1}q_{1}}{2 p_{1}}.\]

• The leader solves

\[\max_{q_{1}} \left\{ p\left(q_{1} + \frac{c - p_{0} - p_{1}q_{1}}{2 p_{1}}\right)q_{1} - c(q_{1}) \right\}.\]

• Its optimality condition is given by

\[\frac{p_{1}}{2}q_{1} + p_{0} + \frac{c - p_{0}}{2} + \frac{p_{1}}{2}q_{1} = c.\]

• Therefore, the leader produces

\[q_{1}^{\ast} = \frac{c - p_{0}}{2 p_{1}}.\]

• Given the leader's quantity, the follower produces

\[q_{2}^{\ast} = b_{2}(q_{1}^{\ast}) = \frac{c - p_{0}}{2 p_{1}} - \frac{1}{2}\frac{c - p_{0}}{2 p_{1}} = \frac{c - p_{0}}{4 p_{1}}.\]

• The market quantity is

\[q^{\ast} = q_{1}^{\ast} + q_{2}^{\ast} = 3\frac{c - p_{0}}{4 p_{1}}.\]

• The market price is

\[p(q^{\ast}) = p_{0} + p_{1} q^{\ast} = \frac{p_{0} + 3 c}{4}.\]

• The follower makes profit

\[\pi_{2} = \left(p(q^{\ast}) - c\right) q_{2}^{\ast} = - \frac{\left(p_{0} - c\right)^{2}}{16p_{1}}.\]

• The leader's profit is

\[\pi_{1} = \left(p(q^{\ast}) - c\right) q_{1}^{\ast} = - \frac{\left(p_{0} - c\right)^{2}}{8p_{1}}.\]

• The firms' profits in the Cournot model with symmetric costs are

\[\pi_{c} = - \frac{\left(p_{0} - c\right)^{2}}{9p_{1}}.\]

• Compared to the firms' profits in the Cournot model, we have

\[\pi_{1} > \pi_{c} > \pi_{2}.\]

• The leader makes more profit compared to Cournot competition, while the follower makes less profit.
• However, aggregate profits in the market are less compared to Cournot, i.e.,

\[\pi_{1} + \pi_{2} < 2 \pi_{c}.\]

• Suppose that firm \(1\) sets a price \(p_{1}\) that is greater than the marginal cost of firm \(2\) (i.e., \(c\)).
• Firm \(2\) can undercut by a small amount and grab all the market. For instance, set price \(p_{2} = \frac{c + p_{1}}{2}\).
• Thus, firm \(1\) can only set a price equal to firm \(2\)'s marginal cost.

• The only possible equilibrium is both firms to set a price equal to the (common) marginal cost.
• Firms do not have any incentive to deviate.
• Setting lower prices leads to losses.
• Setting higher prices leads to zero profits.
• The order with which firms move does not affect the market outcome.

• \(SPE = \left\{ \left(Stay\ out, Fight \right) \right\}\)

• \(SPE = \left\{ \left(Enter, Don't\ fight \right) \right\}\)

• There is an infinite number of potential entrants.
• The market's inverse demand is

\[p(q) = p_{0} + p_{1}q \quad(p_{0}>0,\ p_{1}<0).\]

• All firms have costs

\[c(q) = c_{1}q \quad(c_{1}>0).\]

• The firms that enter the market choose their quantities at the second date simultaneously.
• At the first date, the entrants decide how many of them enter the market.

• When \(n\) firms enter the market, each firm makes profit

\[\pi_{i}(n) = -\frac{(c_{1} - p_{0})^{2}}{(n + 1)^{2} p_{1}} - \bar c.\]

• In equilibrium

\[\pi_{i}(n^{e}) = 0 \iff n^{e} = \frac{p_{0} - c_{1}}{\sqrt{- \bar c p_{1}}} - 1.\]

• The social welfare is calculated by

\[W(n) = \int_{0}^{n} p(s q(s)) \left(q(s) + sq'(s)\right) \mathrm{d} s - n c(q(n)) - n \bar c.\]

• Aggregate profit is maximized when

\[p(n q(n))\left(q(n) + n q'(n)\right) - c(q(n)) - n c'(q(n))q'(n) = \bar c .\]

• This implies that

\[n^{*} = \sqrt[3]{-\frac{(p_{0} - c_{1})^{2}}{p_{1} \bar c}} - 1 .\]

• Comparing with the unregulated entry equilibrium, we have

\[ n^{*} < n^{e} .\]

• With quantity competition, unregulated entry results in excessive entry of firms.

1. Consider an endogenous timing game in which firms simultaneously choose whether to play \(Early\) or to play \(Late\) . If both firms make the same choice, then they compete as in Bertrand competition. If the firms make different choices, then they compete as in the Stackelberg model (sequential competition in quantities), where the leader is the firm that chose \(Early\) and the follower is the firm that chose \(Late\). Suppose that firms have cost functions to \(c(q)=q\), and market demand is given by \(p(q) = 5 - q\).

   1. Calculate the profits of the firms under the different means of competition.
   2. Draw the extensive form of the game.
   3. Calculate all the Nash Equilibria.
   1. If the firms compete simultaneously using prices, they set \(p_{1} = p_{2} = 1\) and make zero profits. If they compete sequentially using quantities, the follower makes profit \[\pi_{f} = \frac{(5-1)^{2}}{16} = 1,\] and the leader makes profit \[\pi_{l} = \frac{(5-1)^{2}}{8} = 2.\]
   2. The extensive form of the game is
   3. There are two pure strategy Nash equilibria, namely \[\left\{\left\{Early, Late\right\}, \left\{Late, Early\right\}\right\},\] and a mixed strategy equilibrium in which each firm chooses \(Early\) with probability \(p=\frac{2}{3}\).

1. Consider the quantity competition with costly entry game. Suppose that firms face market demand \(p(q) = p_{0} + p_{1}q\), have symmetric production costs \(c(q)=c_{1}q\), and entry costs \(\bar c\), where \(p_{0}, c_{1} > 0\) and \(p_{1}<0\). The social welfare as a function of the number of firms is given by \[W(n) = \int_{0}^{n} p(s q(s)) \left(q(s) + sq'(s)\right) \mathrm{d} s - n c(q(n)) - n \bar c .\] Calculate the Pareto optimal number of entrants in the market.

   1. By Fermat's theorem, the welfare is maximized when \(W'(n) = 0\). Using the fundamental theorem of calculus, we get \[p(n q(n))\left(q(n) + n q'(n)\right) - c(q(n)) - n c'(q(n))q'(n) = \bar c.\] The optimal quantity produced by each firm when \(n\) firms enter the market is \[q(n) = -\frac{p_{0} - c_{1}}{(n + 1) p_{1}},\] hence, one calculates \[\frac{q'(n)}{q(n)} = \frac{1}{n + 1}.\] We will use the last expression to simplify the first order condition. Rearranging the necessary condition and using the linearity of costs gives

      \begin{align*} \bar c &= p(n q(n))\left(q(n) + n q'(n)\right) - c(q(n)) - n c'(q(n))q'(n) \\ &= p(n q(n))q(n) - c(q(n)) + n \left(p(n q(n)) - c'(q(n))\right)q'(n) \\ &= \pi(n) + n \left(p(n q(n))q(n) - c(q(n))\right) \frac{q'(n)}{q(n)} \\ &= \pi(n) \left(1 + n \frac{q'(n)}{q(n)}\right) \\ &= \pi(n) \left(1 - n \frac{1}{n + 1}\right) \\ &= \pi(n) \left(\frac{1}{n + 1}\right). \end{align*}

      The profit of each firm when \(n\) firms enter the market is \[\pi(n) = -\frac{(p_{0} - c_{1})^{2}}{(n + 1)^{2} p_{1}},\] so, we get

      \begin{align*} \bar c &= -\frac{(p_{0} - c_{1})^{2}}{(n + 1)^{3} p_{1}}. \end{align*}

      Therefore, the socially optimal number of firms is given by

      \begin{align*} n &= \sqrt[3]{-\frac{(p_{0} - c_{1})^{2}}{p_{1} \bar c}} - 1. \end{align*}

• At the end of the 1980s, the world lysine industry consisted of three significant sellers:
  • Ajinomoto,
  • Kyowa, and
  • Sewon.
• The three largest consumption regions were Japan, Europe, and North America.
• Most production took place in Japan, but it was based on imports of US dextrose.
• Ajinomoto had the largest share of the world market.

• In February 1991, Archer Daniel Midland Co. (ADM) entered the market and built by far the world's largest lysine plant in the US.

• ADM hired biochemist Mark Whitacre, Ph.D., as head of the new division.
• ADM's plant was three times the size of Ajinomoto's largest plant.
• ADM gave Ajinomoto and Kyowa executives an unrestricted tour to show its production capacity.
• Companies engaged in a price war.
  • Three months before ADM's entry, the average US lysine price was \($1.22\) per pound.
  • After an 18-month price war, the US price averaged \($0.68\) per pound.
  • ADM's share of the US market reached \(80\%\).

• After the price war, ADM was willing to soften competition.

• In 1992, Mark Whitacre and his boss Terrance Wilson met with top Ajinomoto and Kyowa managers.
• Wilson proposed forming a world lysine association that would regularly meet.
• The new association would collect and distribute market information.
• Wilson also suggested that the new association could provide a convenient cover for illegal price-fixing discussions! (Connor, 2001)
• After a year, the lysine association was founded, met quarterly, and performed the two functions that Wilson proposed.

WILSON: The only thing we need to talk here because we are gonna get manipulated by these God damn buyers, they're sh, they can be smarter than us if we let them be smarter.

MIMOTO: (Laughs).

WILSON: Okay?

MIMOTO: (ui).

WILSON: They are not your friend. They are not my friend. And we gotta have 'em. Thank God we gotta have 'em, but they are not my friends. You're my friend. I wanna be closer to you than I am to any customer. 'Cause you can make us, I can make money, I can't make money. At least in this kind of a market. And all I wanna is ta tell you again is let's-let's put the prices on the board.

Co-conspirator explains how end-of-year compensation scheme eliminates incentives to cheat on cartel

• In 1992, Mark Whitacre became an FBI whistle-blower. He is the highest-level corporate executive to ever have done so.
• Whitacre's wife pressured him into becoming a whistle-blower. She threatened to inform the FBI herself, if Whitacre wouldn't do it.
• Whitacre informed the FBI that he and other ADM executives performed illegal price-fixing operations.
• Over the next three years, Whitacre collected information and recorded conversations with ADM executives and competitors.
• ADM settlement involved the greatest US federal charges at the time.
• Whitacre was convicted for embezzling and money laundering, which he performed while cooperating with the FBI in the price fixing case.

• Why did ADM initially engage in a price war?

• Why did it initiate the price fixing discussions afterward?
• Why was there so much distrust among companies?

• Suppose that both firms discount future profits by \(\delta \in (0,1)\).
• Firms aim to maximize the series of discounted flow profits, i.e.

\[V_{i} = \max_{(q_{i,t})_{t}} \left\{ \sum_{t=0}^{\infty} \delta^{t} \pi_{i,t}(q_{1,t}, q_{2,t}) \right\}.\]

• There are many equilibria in the game.
• The non coordinating equilibrium is one in which the stage game Nash equilibrium is played at every date.
• All firms use the oligopolistic quantity competition production levels at every date, which leads to flow profits

\[\pi_{i,t} = - \frac{(c_{1}-p_{0})^{2}}{9 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\]

• Each firm's expected future series of profits is

\[V_{i} = - \frac{1}{1-\delta}\frac{(c_{1}-p_{0})^{2}}{9 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\]

• Another equilibrium can be obtained when firms play the trigger strategies.
• Firms produce half the monopolistic quantity and split the monopolistic profits, i.e.,

\[\pi_{i,t} = - \frac{1}{2}\frac{(c_{1}-p_{0})^{2}}{4 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\]

• If firms coordinate with these profits as flow utilities, they get

\[V_{m, i} = - \frac{1}{1-\delta}\frac{(c_{1}-p_{0})^{2}}{8 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\]

• Is there any incentive to deviate?
• Suppose that firm 1 deviates at the current date.
• The best response to firm 2's choice of monopolistic profits is

\[q_{1,0} = \frac{3}{8}\frac{c_{1}-p_{0}}{p_{1}}.\]

• Then, firm 1's profit at the current date is

\[\pi_{1,t} = - \frac{9}{64}\frac{(c_{1}-p_{0})^{2}}{p_{1}}.\]

• On future dates, the firms revert to the non coordinating quantities because they use trigger strategies.
• The total discounted profit of the deviating firm is

\[V_{d,1} = - \frac{9}{64}\frac{(c_{1}-p_{0})^{2}}{p_{1}} - \frac{\delta}{1 - \delta} \frac{(c_{1}-p_{0})^{2}}{9 p_{1}}.\]

• The coordination outcome is sustainable if the deviation profit is less than the coordination profit, namely if

\begin{align*} V_{d,1} &= - \frac{9}{64}\frac{(c_{1}-p_{0})^{2}}{p_{1}} - \frac{\delta}{1 - \delta} \frac{(c_{1}-p_{0})^{2}}{9 p_{1}} \\ &< - \frac{1}{1 - \delta} \frac{(c_{1}-p_{0})^{2}}{8 p_{1}} = V_{m,1}. \end{align*}

• This is equivalent to \(\delta > \frac{9}{17}\).
• As long as firms value future profits enough, there is room for cooperation.

• Suppose again that both discount future profits by \(\delta \in (0,1)\).
• Firms aim to maximize the series of discounted flow profits, i.e.

\[V_{i} = \max_{(p_{i,t})_{t}} \left\{ \sum_{t=0}^{\infty} \delta^{t} \pi_{i,t}(p_{1,t}, p_{2,t}) \right\}.\]

• Firms price at marginal cost when non coordinating.
• They make zero profits at each date.
• Therefore, \(V_{i}=0\).

• As in the case of quantity competition, if firms produce half the monopolistic quantity and split the monopolistic profits, they get

\[\pi_{i,t} = - \frac{1}{2}\frac{(c_{1}-p_{0})^{2}}{4 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\]

• If firms coordinate, then

\[V_{m, i} = - \frac{1}{1-\delta}\frac{(c_{1}-p_{0})^{2}}{8 p_{1}} \quad\quad (i=1,2\quad t\ge 0).\]

• Suppose that firm 1 deviates at the current date.
• The best response to firm 2's choice of monopolistic pricing is to slightly undercut the market price by \(\varepsilon\)
• Then, firm 1 takes all the market and makes almost the monopolistic profit
• For simplicity, drop \(\varepsilon\), so that

\[\pi_{1,t} = - \frac{(c_{1}-p_{0})^{2}}{4 p_{1}}.\]

• On future dates, the firms have zero profits because they use trigger strategies.
• The total discounted profit of the deviating firm is

\[V_{d,1} = - \frac{(c_{1}-p_{0})^{2}}{4 p_{1}}.\]

• The coordination outcome is sustainable if the deviation profit is less than the coordination profit, namely if

  \begin{align*} V_{1,d} &= - \frac{(c_{1}-p_{0})^{2}}{4 p_{1}} \\ &< - \frac{1}{1 - \delta} \frac{(c_{1}-p_{0})^{2}}{8 p_{1}} = V_{1,m}. \end{align*}
• This is equivalent to \(\delta > \frac{1}{2}\).
• As long as firms value future profits enough, there is room for cooperation.
• Why is the lower bound \(\delta\) in quantity competition greater than in price competition?

• The Lerner index (usually denoted \(L\)) is defined as the markup, i.e., the difference between price and marginal cost, as a percentage of the price.

\[L = \frac{p-c'}{p}\]

• Prices are observed, but marginal costs are typically not, and firms are not always eager to reveal this information.

• A concentration index is a statistic that measures the degree of concentration of the market. A market is concentrated when only a few firms have a large share of the market.
• The concentration ratio is the sum of the market shares of a subset of firms in the market. For \(n\) firms in the market, with \(\alpha_{i} = q_{i} / Q\) denoting the share of firm \(i\), the \(m\text{-firms}\) concentration ratio is given by

\[I_{m} = \sum_{i=1}^{m} \alpha_{i}.\]

• If \(I_{m}\) is close to one, it means that most of the market is controlled by the \(m\) firms included in the calculation.
• If \(m\) is small, this suggests that the market is concentrated.

• The Herfindahl Index is a statistic that measures the degree of concentration of the market by considering the full distribution of market shares. It is defined as the sum of squares (\(\mathcal{L}^2\) norm) of market shares of all active firms in the market.
• For \(n\) firms in the market, with \(\alpha_{i} = q_{i} / Q\) denoting the share of firm \(i\), the Herfindahl index is

\[I_{H} = \sum_{i=1}^{n} \alpha_{i}^{2}.\]

• The closer is the value of \(I_{H}\) to one, the more concentrated the market is.

• Suppose that we know the inverse market demand \(p(q,x)\), where \(q\) is the market quantity and \(x\) is a vector of other exogenous characteristics.
• One approach to estimate market power is to start from the equation

\[MR(\lambda) = p + \lambda \frac{\partial p(q,x)}{\partial q} q\]

• If \(\lambda=0\), then \(MR(\lambda) = p\), i.e., the market is perfectly competitive.
• If \(\lambda=1\), then the market is a monopoly, i.e.,

\[MR(\lambda) = p + \frac{\partial p(q,x)}{\partial q} q.\]

• If \(\lambda=1/n\), then the market is represented by a symmetric \(n\text{-firm}\) Cournot model, i.e.,

\[MR(\lambda) = p + \frac{1}{n}\frac{\partial p(q,x)}{\partial q} q.\]

• Suppose that \(c'(q,w)\) is the marginal cost function, where \(w\) is a vector of exogenous characteristics affecting cost.
• In equilibrium, for all firms in the market, it holds

\[p + \lambda \frac{\partial p(q,x)}{\partial q} q = c'(q,w).\]

• From this expression, we can obtain estimates of \(\lambda\).

1. Consider two firms competing in quantities for an infinite number of dates. The discount factor is \(\delta \in (0, 1)\).The market's inverse demand is \(p(q)=220-q\), and both firms have a marginal cost equal to \(10\)

   1. Find the equilibrium under non cooperation (Cournot competition) at each date. Calculate the quantity that a monopolist would produce at each date. Calculate the profits in both cases and compare them.
   2. Write down the trigger strategies of the two firms that can lead to tacit collusion.
   3. What is the most profitable deviation from the trigger strategy at the initial date? What is the profit of the deviation at this date? How does the profit compare with the coordination profit of the trigger strategy?
   4. For which values of \(\delta\) is collusion sustainable?

   1. In the stage game, each firm maximizes \[\max_{q_{i}} (220 - q_{i} - q_{j} - 10 ) q_{i},\] which implies that \(q_{i} = q_{j} = q_{c} = 70\). When firms do not cooperate, they choose to produce these quantities at all dates. In this case, the market price is \[p(2q_{c}) = 220 - 140 = 80,\] and each firm makes profit \[\pi_{c} = (80 - 10 ) 70 = 4900.\]

      The monopolist would instead maximize \[\max_{q} (220 - q - 10 ) q,\] which gives \(q_{m} = 105\). The market price is then \[p(q_{m}) = 220 - 105 = 115,\] and the monopolist makes profit \[\pi_{m} = (115 - 10) 105 = 11025.\]

   2. The trigger strategy that can lead to collusion is for each firm \(i\) to produce according to the rule

      \begin{align*} q_{i,t} = \left\{ \begin{aligned} \frac{q_{m}}{2} & \quad\quad q_{j,s}=\frac{q_{m}}{2} \text{ for all } s < t \\ q_{c} & \quad\quad \text{ otherwise } \end{aligned}\right. . \end{align*}
   3. Given that firm \(j\) plays the trigger strategy, firm \(i\) maximizes \[\max_{q_{i}} \left(220 - q_{i} - \frac{105}{2} - 10\right) q_{i},\] which implies that the best firm \(i\) can do, if it deviates from the trigger strategy, is to produce \(q_{d} = \frac{315}{4}\). The profit of firm \(i\) is then \[\pi_{d} = \frac{99225}{16},\] which is greater than \(pi_{m}/2\). Thus, the firm can make greater flow profit at the initial date if it deviates from coordination.
   4. This practice, however, does not make the firm overall better off. Only short-sighted firms prefer this scenario. Firms that value future profits prefer coordination. Specifically, coordination can be sustained if and only if \[\frac{1}{1-\delta} \frac{\pi_{m}}{2} \ge \pi_{d} + \frac{\delta}{1 -\delta}\pi_{c}\] or equivalently \[\delta \ge \frac{\frac{\pi_{m}}{2} - \pi_{d}}{\pi_{c}- \pi_{d}} = \frac{\frac{11025}{2} - \frac{99225}{16}}{4900 - \frac{99225}{16}} = \frac{11025}{20825} = \frac{9}{17}.\]

2. Consider two firms competing in prices for a potentially infinite number of dates. The two firms have marginal costs equal to \(c\). If the two firms set the same price, they split the market so that firm \(1\) gets a share \(\mu\) and firm \(2\) get a share \(1-\mu\). The firms do not discount future profits, but there is a probability \(p > 0\) that the market closes down after each date. The firms use trigger strategies to coordinate and split the monopolistic profit.

   1. Find the equilibrium if \(p=1\).
   2. Suppose that \(\mu = \frac{1}{2}\). Find the values of \(p<1\) for which collusion is sustainable. How do you interpret the result?
   3. Find the values of \(p<1\) for which collusion is sustainable for any value of \(\mu\in(0,1)\). What is the equilibrium relationship between \(p\) and \(\mu\)? How do you interpret the result?
   1. If \(p=1\), the firms compete only for one date, and the game reduces to Bertrand competition. The firms set \(p_{1}=p_{2}=c\).
   2. The most profitable deviation of each firm at the initial date is to slightly undercut the monopoly price by \(\varepsilon\). By doing so, the firm can take all the market demand and achieve almost the monopolistic profit. For simplicity, assume that the deviating firm earns exactly the monopolistic profit. Then, the deviation is not profitable if and only if \[\frac{1}{p}\frac{\pi_{m}}{2} \ge \pi_{m} \iff p \le \frac{1}{2}.\] Collusion is sustainable if the probability that the market closes down is small and the firms have the potential to benefit from future profits.
   3. With arguments analogous to the \(\mu=\frac{1}{2}\) case, the condition for firm \(1\) is \[\frac{1}{p}\mu \pi_{m} \ge \pi_{m} \iff p \le \mu,\] and for firm \(2\) is \[\frac{1}{p}(1-\mu) \pi_{m} \ge \pi_{m} \iff p \le 1-\mu.\] Therefore, collusion is sustainable if and only if \[p \le \min\{\mu, 1-\mu\}.\] If \(\mu < \frac{1}{2}\), the equilibrium condition becomes less stringent when \(\mu\) increases. In contrast, the equilibrium condition becomes more stringent when \(\mu\) increases for \(\mu>\frac{1}{2}\). The most lenient equilibrium condition is obtained for \(\mu=\frac{1}{2}\). The further is \(\mu\) from \(\frac{1}{2}\), the less are the flow profits that one of the two firms makes and, as a result, coordination based on future benefits becomes less profitable.

1. Consider two firms competing in quantities for an infinite number of dates. The discount factor is \(\delta \in (0, 1)\).The market's inverse demand is \(p(q)=p_{0}+p_{1}q\) and both firms have marginal cost equal to \(c\), where \(p_{0}>c>0\) and \(p_{1}<0\). Suppose that firms use the following tit-for-tat strategy. If the other firm has produced half the monopolistic quantity last date, then produce half the monopolistic quantity at this date. Otherwise, produce the Cournot quantity.

   1. Find the equilibrium under non cooperation (Cournot competition) at each date. Calculate the profits of the firms.
   2. Find the total discounted profits if each firm produces half the monopolistic quantity at each date.
   3. Find the most profitable deviation of firm \(i\) at the initial date. Calculate the profit of the deviating firm at the initial date.
   4. Suppose that at the second date, the deviating firm reverts to producing half the monopolistic quantity. Calculate the profit it makes at this date.
   5. For which values of \(\delta\) is collusion sustainable?
   1. As in the grim trigger strategy case, if the firms do not coordinate, they produce \[q_{c} = \frac{c - p_{0}}{3 p_{1}},\] make flow profits \[\pi_{c} = -\frac{(p_{0} - c)^{2}}{9 p_{1}},\] and total discounted profits \[V_{c} = -\frac{1}{1-\delta}\frac{(p_{0} - c)^{2}}{9 p_{1}}.\]
   2. In this case, each firm produces \[\frac{q_{m}}{2} = \frac{c - p_{0}}{4 p_{1}},\] makes flow profit \[\frac{\pi_{m}}{2} = -\frac{(p_{0} - c)^{2}}{8 p_{1}},\] and total discounted profit \[V_{m} = -\frac{1}{1-\delta}\frac{(p_{0} - c)^{2}}{8 p_{1}}.\]
   3. The most profitable deviation of firm \(i\) is calculated by solving \[\max_{q_{i}} \left(p_{0} + p_{1} \left(q_{i} + \frac{q_{m}}{2}\right) - c\right) q_{i},\] which leads to \[q_{1} = \frac{3}{8}\frac{c-p_{0}}{p_{1}},\] and flow profit \[\pi_{1} = -\frac{9}{64}\frac{(c-p_{0})^{2}}{p_{1}}.\]

   4. Firm \(i\) is producing half the monopolistic quantity, and firm \(j\) is producing the Cournot quantity, punishing the initial date deviation. Thus, firm \(i\) makes profit

      \begin{align*} \pi_{2} &= \left(p_{0} + p_{1} \left(\frac{q_{m}}{2} + q_{c}\right) - c\right) \frac{q_{m}}{2} \\ &= -\frac{5}{48}\frac{(c-p_{0})^{2}}{p_{1}}. \end{align*}

   5. There are two cases we should consider. The deviating firm may revert back to producing half the monopolistic quantity after the deviation, or it can resort to producing the Cournot quantity. In the first case, the payoff is \[V_{r} = \pi_{1} + \delta \pi_{2} + \delta^{2} V_{m},\] while in the second case, it is \[V_{n} = \pi_{1} + \delta V_{c}.\] The second case is excluded if and only if

      \begin{align*} V_{r} \ge V_{n} &\iff \pi_{2} + \delta V_{m} \ge V_{c} \\ &\iff -\frac{5}{48}\frac{(c-p_{0})^{2}}{p_{1}} - \frac{\delta}{1-\delta} \frac{1}{8}\frac{(c-p_{0})^{2}}{p_{1}} \ge - \frac{1}{1-\delta} \frac{1}{9}\frac{(c-p_{0})^{2}}{p_{1}} \\ &\iff \delta \ge \frac{1}{3}. \end{align*}

      Moreover, for collusion to be sustainable, we need

      \begin{align*} V_{m} \ge V_{r} &\iff V_{m} \ge \pi_{1} + \delta \pi_{2} + \delta^{2} V_{m} \\ &\iff -\frac{1 - \delta^{2}}{1-\delta} \frac{1}{8} \frac{(c-p_{0})^{2}}{p_{1}} \ge -\frac{9}{64}\frac{(c-p_{0})^{2}}{p_{1}} - \delta \frac{5}{48}\frac{(c-p_{0})^{2}}{p_{1}} \\ &\iff (1 + \delta) \frac{1}{8} \ge \frac{9}{64} + \delta \frac{5}{48} \\ &\iff \delta \ge \frac{3}{4}. \end{align*}

• Stay away from utilities!

• Railways are affordable but unlikely to get you anywhere fast.
• When playing against one opponent, orange or light blue sets are good.
• Also good with more opponents, but only in the early stages.
• With two or three opponents (longer game), orange and red are better.
• With more than three opponents, green becomes your best shot.

• An imperfect information game is a game in which at least one player is uninformed about the actions of other players.
• All one shot (static) games in which players move simultaneously are imperfect information games. E.g., the prisoners' dilemma.
• Sequential games with non trivial information sets (i.e., sets that contain more than one decision node) are imperfect information games.

• An incomplete information game is a game in which at least one player is unaware of the types of other players.
• For example, participants in auctions are not fully informed about other participants' preferences (valuations) for the auctioned objects.
• In bargaining, players may not know the reservation values of other players.

• The Harsanyi's transformation is a method that allows us to represent static Bayesian games in the extensive form.
• The idea of the transformation is to include an additional player (nature) to represent various states of the game.
• The Bayesian extensive form of a Bayesian game is the extensive form representation resulting from Harsanyi's transformation.

• What is the Bayesian extensive form of the previous Bayesian game example?

• Consolidating the players' choices under different types and calculating the expected payoffs allows us to represent static Bayesian games in the normal form.
  • Each player's actions are combinations of actions at each state of the game.
  • The payoffs of each player are expected values over the combined actions.
• The Bayesian normal form is the consolidating table form representation of a Bayesian game.

• How can we find the strategies in the Bayesian normal form of the example?
• Player \(A\) can be in state \(x=12\) or in state \(x=0\). Thus, she has actions \[\left\{Top_{12}Top_{0}, Top_{12}Bottom_{0}, Bottom_{12}Top_{0}, Bottom_{12}Bottom_{0}\right\}.\]
• Player \(B\) has only one type, so her actions are \(\left\{Left, Right\right\}\).

• How can we find the payoffs in the Bayesian normal form of the example?

• The expected payoffs when \(A\) chooses \(Top_{12}Top_{0}\) and \(B\) chooses \(Left\) are

  \begin{align*} \mathbb{E} u_{A}(Top_{12}Top_{0}, Left) &= \frac{2}{3}12 + \frac{1}{3}0 = 8 \\ \mathbb{E} u_{B}(Top_{12}Top_{0}, Left) &= \frac{2}{3}9 + \frac{1}{3}9 = 9 \end{align*}

• The expected payoffs when \(A\) chooses \(Top_{12}Bottom_{0}\) and \(B\) chooses \(Left\) are

  \begin{align*} \mathbb{E} u_{A}(Top_{12}Bottom_{0}, Left) &= \frac{2}{3}12 + \frac{1}{3}6 = 10 \\ \mathbb{E} u_{B}(Top_{12}Bottom_{0}, Left) &= \frac{2}{3}9 + \frac{1}{3}0 = 6 \end{align*}

• What is the Bayesian normal form of the previous Bayesian game example?

• What is the Bayesian Nash equilibrium of the previous Bayesian game example?

• \(BNE = \left\{ \left\{Bottom_{12}Bottom_{0}, Right\right\} \right\}\)

• Consider the interaction of a seller and buyer in the used automobile market.
• The quality of the car is stochastic.
• Nature draws a peach (car of good quality) with probability \(q\) and a lemon (car of bad quality) with probability \(1-q\).
• The seller has more information than the buyer.
• Before she plays, she observes whether she got a peach or a lemon.
• The buyer does not have this information (information asymmetry). She only knows the distribution of types (\(q\), \(1-q\)).

• Suppose that irrespective of the type, the buyer always values the car more than the seller (so trade can be beneficial).
  • A peach is worth \(3\) for the buyer and \(2\) for the seller.
  • A lemon is worth \(1\) for the buyer and \(0\) for the seller.
• Both players simultaneously decide whether to trade or not at a market price \(p\).
• However, if the market price \(p\) is above the buyer's valuation for a lemon, the buyer would not want to trade a lemon at the market price.

• The Bayesian normal form of the game is

• The no trade equilibrium \(\left\{NN, N\right\}\) is valid for all prices \(p\ge 0\).
• Additionally, if \(p\ge 3\) (car is too expensive), then there are three more BNE in which no trade occurs.
  • The buyer always does not trade, and the seller can choose any action \(\left\{ \left\{TT, N\right\}, \left\{TN, N\right\}, \left\{NT, N\right\} \right\}\).
• Instead, if \(1\le p \le 3\), there are two more no trade equilibria.
  • The seller wants to trade only lemons. The buyer does not trade because the price is too high for a lemon \(\left\{NT, N\right\}\).
  • The seller always trades, but the probability of having peach is small, i.e., \(q\le (p-1)/2\).
  • The buyer thinks that it is more likely to pay too high a price for a lemon, so she does not trade \(\left\{TT, N\right\}\).

• If \(0\le p\le 1\) (car is too cheap), then there is a BNE in which only lemons are traded.
  • The price does not cover the peach valuation of the seller.
  • The seller only trades lemons. The buyer agrees because the price is low \(\left\{NT, T\right\}\).

• If \(2\le p\le 3\), there is a BNE in which both lemons and peaches are traded.
  • The seller always trades, and the probability of having peach is good, i.e., \(q\ge (p-1)/2\).
  • The buyer thinks that it is quite likely to pay a good price for a peach, so she trades \(\left\{TT, T\right\}\).

1. Consider the following static game of incomplete information. Nature selects the type of player \(A\), where \(x = 2\) with probability \(2/3\) and \(x = 0\) with probability \(1/3\). Player \(A\) observes her type, but player \(B\) does not. Players choose their actions simultaneously and independently.

   1. Draw the Bayesian normal form.
   2. Find the Bayesian Nash equilibria.
   1. The Bayesian normal form is
   2. The unique Bayesian Nash equilibrium of the game is \(\left\{BottomTop, Right\right\}\).

2. Consider a second-price sealed-bid auction. Two potential buyers (buyer \(1\) and buyer \(2\)) simultaneously and independently submit sealed bids \(b_{1}\) and \(b_{2}\) for buying an auctioned object. The object is awarded to the highest bidder. However, the price of the object is not set equal to the highest bid (winning bid), but equal to the second highest bid (losing bid). If the two bids are equal, then player \(i\) wins the auction with probability \(1/2\) and pays the common bid. Buyer \(i=1,2\) has a personal valuation \(v_{i}>0\) for the auctioned object. The valuations are private information, but their distribution is common knowledge.

   1. Find the Bayesian Nash equilibrium of the game.
   2. Is the equilibrium economically efficient?

   1. We will show that setting a bid \(b_{i}^{\ast}=v_{i}\) weakly dominates any other bidding strategy. That is, we will show that bidding strategies \(b_{i}\neq v_{i}\) do not give greater payoffs than \(b_{i}^{\ast}=v_{i}\). Suppose that player \(i\) has valuation \(v_{i}\) and considers a bid \(b_{i}\). We examine the following two alternative cases.

      Firstly, consider \(b_{i}>v_{i}\). There are four potential outcomes for this bid. Firstly, if the other player's bid is greater than \(b_{i}\), i.e., \(b_{j}>b_{i}\), then both \(b_{i}\) and \(b_{i}^{\ast}\) result in zero payoffs for player \(i\). Secondly, if the two bids are equal, then player \(i\)'s expected payoff is \(\frac{1}{2}(v_{i}-b_{i})<0\). Thus, player \(i\) is doing better by deviating to \(b_{i}^{\ast}\). Thirdly, if the other player's bid is between \(b_{i}\) and \(v_{i}\), i.e., \(b_{i}>b_{j}>v_{i}\), then player \(i\) wins the object and pays a price \(b_{j}\). Her payoff is then \(v_{i}-b_{j}<0\), and the player can do better (zero payoff) if she deviates to \(b_{i}^{\ast}\). Fourthly, if the other player's bid is less than player \(i\)'s valuation, i.e., \(v_{i} \ge b_{j}\), then player \(i\) wins the object and has a payoff \(v_{i}-b_{j}\ge 0\). Her payoff is the same if she bids \(b_{i}^{\ast}\).

      Secondly, consider \(b_{i} < v_{i}\). Due to symmetry, the previous paragraph implies that we should focus to cases for which \(b_{j}\le v_{i}\). Firstly, if \(b_{j}=v_{i}\), player \(i\) has zero payoff. Deviating to \(b_{i}^{\ast}\) gives payoff \(\frac{1}{2}(v_{i}-b_{i})=0\). Thus, the player is indifferent between \(b_{i}\) and \(b_{i}^{\ast}\). Secondly, if \(b_{i} < b_{j} < v_{i}\), player \(i\) has zero payoff. Deviating to \(b_{i}^{\ast}\) gives payoff \(v_{i}-b_{j} > 0\), which makes player \(i\) better off. Thirdly, if \(b_{i}=b_{j} < v_{i}\), player \(i\) has expected payoff \(\frac{1}{2}(v_{i}-b_{i}) > 0\). Her payoff is doubled if she deviates to \(b_{i}^{\ast}\). Finally, if \(b_{j} < b_{i} < v_{i}\), player \(i\) has expected payoff \(v_{i}-b_{j} > 0\). Player \(i\) achieves the same payoff by deviating to \(b_{i}^{\ast}\).

   2. Yes, the auction mechanism allocates the object to the buyer with the highest valuation.

1. Consider a first-price sealed-bid auction. Two potential buyers (buyer \(1\) and buyer \(2\)) simultaneously and independently submit sealed bids \(b_{1}\) and \(b_{2}\) for buying an auctioned object. The object is awarded to the highest bidder. The price of the object is equal to the winning bid. Buyer \(i=1,2\) has a personal valuation \(v_{i}\) for the auctioned object, which is independently, identically, and uniformly distributed on \([0,1]\). The valuations are private information.

   1. Find the Bayesian Nash equilibrium of the game [Hint: guess that optimal bids are linear in buyers' valuations].
   2. Is the equilibrium economically efficient?

   1. We will show that the auction has a symmetric equilibrium of the form \(b_{i}^{\ast}(v)=\alpha v\) using the method of undetermined coefficients. Buyer \(i\) maximizes her expected payoff \[u_{i} = \max_{b_{i}} \mathbb{E} (v_{i} - b_{i}) = \max_{b_{i}} (v_{i} - b_{i}) \mathbb{P}(b_{i} \ge b_{j}).\] The bid equality has zero probability and can be ignored. Thus,

      \begin{align*} u_{i} &= \max_{b_{i}} (v_{i} - b_{i}) \mathbb{P}(b_{i} > \alpha_{j} v_{j}) \\ &= \max_{b_{i}} (v_{i} - b_{i}) \frac{b_{i}}{\alpha}. \end{align*}

      The necessary condition is \[2 b_{i} = v_{i},\] which implies that \(\alpha=1/2\). The objective is strictly concave, so the solution represents a unique maximum.

   2. Yes, the auction mechanism allocates the object to the buyer with the highest valuation.

• The business model for the PageRank tool was also uncertain.

• With catalog search, Yahoo could place relevant ads in the corresponding catalogs.
• The PageRank tool would take users away from searching the catalog and reduce Yahoo's advertising revenue.
• Yahoo was not interested in PageRank.
• Soon after, PageRank was patented and became the core algorithm of Google's search engine.
• Google introduced algorithms associating search keywords with relevant ads.
• Instead of reducing advertising revenue, the alternative approach led to much greater revenues than Yahoo ever achieved.

• The profit of firm \(i\) is stochastic, even after \(c_{s,i}\) is observed, because the cost of firm \(j\) is not observed.
• Given the observed value of \(c_{s,i}\), firm \(i\) maximizes its expected profit.

\[\max_{q_{s,i}} \sum_{r=l,h} \frac{1}{2} \left[ p(q_{s,i} + q_{r,j})q_{s,i} - c(q_{s,i}, s)\right] \quad\quad (s=l,h).\]

• The necessary condition for firm \(i\) is

\[p_{0} + 2 p_{1} q_{s,i} + p_{1} \sum_{r=l,h} \frac{1}{2} q_{r,j} = c_{s,i}.\]

• Solving for \(q_{s,i}\) gives the best response of firm \(i\) at state \(c_{s,i}\)

\[q_{s,i} = \frac{c_{s,i} - p_{0} - p_{1} \sum_{r=l,h} \frac{1}{2} q_{r,j}}{2 p_{1}}.\]

• Calculating the expected value of the best responses of \(i\) gives

\[\sum_{s=l,h} \frac{1}{2} q_{s,i} = \frac{\sum_{s=l,h} \frac{1}{2} c_{s,i} - p_{1} \sum_{r=l,h} \frac{1}{2} q_{r,j} - p_{0}}{2 p_{1}}.\]

• Since the game is symmetric, we expect \(q_{s,j}=q_{s,i}\) for both states \(s=l,h\), so

\[\sum_{s=l,h} \frac{1}{2} q_{s,i} = \frac{\sum_{s=l,h} \frac{1}{2} c_{s,i} - p_{0}}{3 p_{1}}.\]

• Substituting into the best response, we obtain

\[q_{s,i} = \frac{3c_{s,i} - 2 p_{0} - \sum_{r=l,h} \frac{1}{2} c_{r,i}}{6 p_{1}} = \frac{3c_{s,i} - 2 p_{0} - \mathbb{E} c_{i}}{6 p_{1}}.\]

• Firm \(i\) may produce less or more compared to the deterministic case (say \(q\)) depending on the realization of \(c_{s,i}\).
  • If \(c_{s,i}>\mathbb{E} c_{i}\), then production is too costly and \(q_{s,i} < q\).
  • If \(c_{s,i}<\mathbb{E} c_{i}\), then production is cheap and \(q_{s,i}>q\).
  • On average, uncertainty cancels out \(\mathbb{E} q_{i} = q\).
• The market price and quantity can be either greater or less than in the deterministic case, with uncertainty also canceling out on average.
• However, uncertainty reduces the profits of the firms (why?).
  • The more uncertain production gets (i.e., the more volatile is \(c_{s,i}\)), the fewer profits firms make on average.

• Given the choice \(p_{j}\) of the other firm, firm \(i\) maximizes its expected profit

  \begin{align*} \pi_{i} &= \max_{p_{i}} \mathbb{E} \left[ \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(p_{i}(\theta_{i}), p_{j}(\theta_{j}))\right] \\ &= \max_{p_{i}} \left\{ \left( p_{i}(\theta_{i}) - \theta_{i} \right) (1 - p_{i}(\theta_{i})) \mathbb{P}(p_{i}(\theta_{i}) < p_{j}(\theta_{j}) ) \right\} . \end{align*}

• With a bit more mathematics and some calculations…
• Firms choose \[p^{\ast}(\theta_{i}) = \frac{1 + 2 \theta_{i}}{3} \quad (i =1, 2).\]
• In contrast with the deterministic case, all firms price above their marginal cost, i.e., \[p^{\ast}(\theta_{i}) > \theta_{i}.\]
• For \(n\) firms the equilibrium prices are \[p^{\ast}(\theta_{i}) = \frac{1 + n \theta_{i}}{n+1} \quad (i =1, \dots, n).\]
• The markup decreases in the number of firms in the market (the market becomes more competitive) \[\frac{\mathrm{d} p^{\ast}(\theta_{i})}{\mathrm{d} n} < 0.\]
• The limiting case gives the perfect competition pricing \[\lim_{n\to\infty} p^{\ast}(\theta_{i}) = \theta_{i}.\]

Consider, again, the Bayesian Nash equilibrium of competition in quantities when firms have symmetric costs, i.e., \[q_{s,i} = \frac{3c_{s,i} - 2 p_{0} - \mathbb{E}c_{i}}{6 p_{1}}.\] We can rewrite the optimal quantity as \[q_{s,i} = \underbrace{\frac{\mathbb{E}c_{i} - p_{0}}{3 p_{1}}}_{:=q} + \underbrace{\frac{c_{s,i} - \mathbb{E}c_{i}}{2 p_{1}}}_{:=u_{s}}.\] The first part on the right hand side of the last expression (i.e., \(q\)) corresponds to the optimal quantity of the model with deterministic costs. From the second term (\(u_{s}\)), we see that if the realized shock (\(c_{s,i}\)) is greater than the average shock, namely \(c_{s,i}-\mathbb{E}c_{i}>0\), then the firm produces less than the in deterministic case (because \(p_{1}<0\)). Analogously, when the realized shock is less than the mean shock, the second part becomes positive, and firm \(i\) produces more compared to the deterministic case. On average, these differences cancel out, as we can see from \[\mathbb{E} q_{i} = \frac{3\mathbb{E} c_{i} - 2 p_{0} - \mathbb{E}c_{i}}{6 p_{1}} = \frac{\mathbb{E}c_{i} - p_{0}}{3 p_{1}}.\] The interpretation is that when the marginal production cost of a firm unexpectedly increases, the firm reacts by reducing the quantity it produces.

The market quantity depends on the realization of both firms' shock. Specifically, it is \[Q_{s,r} = q_{s,i} + q_{r,j} = 2\frac{\mathbb{E}c_{i} - p_{0}}{3 p_{1}} + \frac{c_{s,i} + c_{r,j} - 2\mathbb{E}c_{i}}{2 p_{1}}.\] Depending on the realizations of \(c_{s,i}\) and \(c_{r,j}\), the market quantity can be greater or less than when costs are deterministic. Nevertheless, on average, we get \[\mathbb{E} Q_{s,r} = 2\frac{\mathbb{E}c_{i} - p_{0}}{3 p_{1}},\] similar to the case without uncertainty. A analogous conclusion is true for the market price, which is given by \[P_{s,r} = p_{0} + p_{1}Q_{s,r} = \frac{2\mathbb{E}c_{i} + p_{0}}{3} + \frac{c_{s,i} + c_{r,j} - 2\mathbb{E}c_{i}}{2}.\] In the stochastic case, prices can be above or below the deterministic price level, but the uncertainty cancels out on average.

The profits of the firm are a bit more tedious to calculate. Specifically,

\begin{align*} \pi_{s,r,i} &= \left(p_{0} + p_{1}Q_{s,r} - c_{s,i}\right) q_{s,i} \\ &= \left(p_{0} + p_{1}\left(2 q + u_{s} + u_{r}\right) - c_{s,i}\right) \left(q + u_{s}\right) \\ &= \left(p_{0} + p_{1}2 q - \mathbb{E}c_{i}\right) q + \left(p_{1}\left(u_{s} + u_{r}\right) + \mathbb{E}c_{i} - c_{s,i}\right) q \\ &\qquad + \left(p_{0} + 2p_{1} q - \mathbb{E}c_{i}\right) u_{s} + \left(p_{1}\left(u_{s} + u_{r}\right) + \mathbb{E}c_{i} - c_{s,i}\right) u_{s}. \end{align*}

Since \(u_{s}\), \(u_{r}\) are independent and \(\mathbb{E}u_{s}=0=\mathbb{E}u_{r}\), expected profits are easily calculated from the last expression as

\begin{align*} \mathbb{E} \pi_{i} &= \left(p_{0} + p_{1}2 q - \mathbb{E}c_{i}\right) q + p_{1}\mathbb{E} u_{s}^2 + \mathbb{E}\left[\left(\mathbb{E}c_{i} - c_{s,i}\right) u_{s}\right] \\ &= \left(p_{0} + p_{1}2 q - \mathbb{E}c_{i}\right) q + p_{1}\mathbb{E} u_{s}^2 -2p_{1} \mathbb{E} u_{s}^2 \\ &= -\frac{(p_{0}-c_{1})^{2}}{9p_{1}} - p_{1}\mathbb{E} u_{s}^2 \\ &= -\frac{(p_{0}-c_{1})^{2}}{9p_{1}} - p_{1}\mathbb{V} u_{s} \\ &= -\frac{(p_{0}-c_{1})^{2}}{9p_{1}} - \frac{1}{4p_{1}}\mathbb{V} c_{i}. \end{align*}

Because profits are quadratic in marginal production costs, they are affected by the uncertainty in production. The more uncertain production gets (i.e., the greater the variance of \(c_{s,i}\) becomes), the fewer profits firms make on average.

Suppose that there are \(n\) firms in the market. Firm \(i=1,2,\dots,n\) has cost function \[c(q_{i}, \theta_{i}) = \theta_{i} q_{i},\] where the random variables \(\theta_{i}\) are identically, independently, and uniformly distributed on \([0, 1]\). The value of \(\theta_{i}\) is observed by firm \(i\) but not by other firms (private information).

Let the demand for firm \(i\) be

\begin{align*} d_{i}(p_{1}, \dots, p_{n}) = \left\{\begin{aligned} &1 - p_{i} & p_{i} < \min_{j\neq i} p_{j} \\ &\frac{1 - p_{i}}{m} & p_{i} = \min_{j\neq i} p_{j},\ m=\left|\operatorname{arg}\min_{j\neq i} p_{j}\right| \\ &0 & p_{i} > \min_{j\neq i} p_{j} \end{aligned}\right.. \end{align*}

How do Symmetric Bayesian Nash Equilibria Look Like? The Bayesian Nash Equilibria of the game specify best responses (prices) for each state (\(\theta\)). Thus, we look for functions \(p_{i}^{\ast}\) giving a price for each \(\theta\in[0,1]\). Due to symmetry, we expect \(p^{\ast} = p_{i}^{\ast}\) for every firm \(i\). The probability of two or more firms having the same costs is zero. So, we can focus on firms with different costs. Additionally, We expect \(p^{\ast}(1)=1\) because prices below cost (\(p^{\ast}(1)<\theta=1\)) or above the market size (\(p^{\ast}(1)>d_{i}(0)\)) give negative profit. From our intuition in the deterministic case, we expect \(p^{\ast}\) to be increasing in the marginal cost \(\theta\).

We focus on smooth best response functions \(p^{\ast}\). Given the choices \(p_{j}\) of other firms, firm \(i\) maximizes expected profits

\begin{align*} \pi_{i} &= \max_{p_{i}} \mathbb{E} \left[ \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(p_{1}(\theta_{1}), \dots, p_{i}(\theta_{i}), \dots, p_{n}(\theta_{n}))\right] \\ &= \max_{p_{i}} \left\{ \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(\dots) \mathbb{P}(p_{i}(\theta_{i}) < \min_{j\neq i} p_{j}(\theta_{j}) ) \right\} \\ &= \max_{p_{i}} \left\{ \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(\dots) \prod_{j\neq i} \mathbb{P}(p_{i}(\theta_{i}) < p_{j}(\theta_{j}) ) \right\} \\ &= \max_{p_{i}} \left\{ \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(\dots) \prod_{j\neq i} \mathbb{P}(p_{j}^{-1}(p_{i}(\theta_{i})) < \theta_{j} ) \right\} \\ &= \max_{p_{i}} \left\{ \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(\dots) \prod_{j\neq i} (1 - p_{j}^{-1}(p_{i}(\theta_{i}))) \right\} . \end{align*}

To find the necessary conditions, by Pontryagin's maximum principle, we can maximize pointwise. We then get

\begin{align*} & (1 - 2p_{i}(\theta_{i}) + \theta_{i}) \prod_{j\neq i} (1 - p_{j}^{-1}(p_{i}(\theta_{i}))) - \\ & \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(\dots) \sum_{k \neq i} \frac{\mathrm{d} p_{j}^{-1}(p_{i}(\theta_{i}))}{\mathrm{d} p_{i}(\theta_{i})} \prod_{j\neq k, i} (1 - p_{j}^{-1}(p_{i}(\theta_{i}))) = 0 . \end{align*}

By the inverse mapping theorem

\begin{align*} & (1 - 2p_{i}(\theta_{i}) + \theta_{i}) \prod_{j\neq i} (1 - p_{j}^{-1}(p_{i}(\theta_{i}))) - \\ & \left( p_{i}(\theta_{i}) - \theta_{i} \right) d_{i}(\dots) \sum_{k \neq i} \frac{1}{p_{j}'(p_{j}^{-1}(p_{i}(\theta_{i})))} \prod_{j\neq k, i} (1 - p_{j}^{-1}(p_{i}(\theta_{i}))) = 0. \end{align*}

By symmetry, we expect \(p_{i} = p_{j} = p^{\ast}\), so we can simplify to

\begin{align*} & (1 - 2p^{\ast}(\theta_{i}) + \theta_{i}) (1 - \theta_{i})^{n-1} - \\ & \left( p^{\ast}(\theta_{i}) - \theta_{i} \right) d_{i}(\dots) (n - 1) \frac{1}{(p^{\ast})'(\theta_{i})} (1 - \theta_{i})^{n-2} = 0. \end{align*}

Thus, the necessary condition specifies a first order differential equation that \(p^{\ast}\) needs to satisfy.

The best responses are obtained by solving the differential equation. Rearranging gives

\begin{align*} (p^{\ast})'(\theta_{i}) &= \frac{\left( p^{\ast}(\theta_{i}) - \theta_{i} \right) (1 - p_{i}(\theta_{i})) (n - 1)}{(1 - 2p^{\ast}(\theta_{i}) + \theta_{i}) (1 - \theta_{i})}. \end{align*}

We solve using the method of undetermined coefficients for \(p^{\ast}(\theta_{i}) = p_{0} + p_{1}\theta_{i}\). By \(1=p(1)\), this simplifies to \(p^{\ast}(\theta_{i}) = 1 - p_{1} + p_{1}\theta_{i}\). Substituting gives

\begin{align*} p_{1} &= \frac{(1 - p_{1}) p_{1} ( 1 - \theta_{i})^{2} (n - 1)}{(2 p_{1} - 1)(1 - \theta_{i})^{2}}. \end{align*}

Thus,

\begin{align*} p_{1} = \frac{n}{n+1} \quad\text{and}\quad p_{0} = \frac{1}{n+1}. \end{align*}

In the symmetric Bayesian Nash equilibrium firms choose \[p^{\ast}(\theta_{i}) = \frac{1 + n \theta_{i}}{n+1} \quad (i =1, \dots, n).\] In contrast with the deterministic case, all firms price above their marginal cost, i.e., \[p^{\ast}(\theta_{i}) > \theta_{i}.\] The markup decreases in the number of firms in the market (the market becomes more competitive) \[\frac{\mathrm{d} p^{\ast}(\theta_{i})}{\mathrm{d} n} < 0.\] The limiting case gives the perfect competition pricing \[\lim_{n\to\infty} p^{\ast}(\theta_{i}) = \theta_{i}.\]

1. Consider a Cournot duopoly with information asymmetry. The market inverse demand is \(p(q) = 10 - q\). The cost function of firm \(1\) is \(c(q)=q\), which is common knowledge. The cost of firm two is \(c(q)=0\) with probability \(1/2\) and \(c(q)=2 q\) with probability \(1/2\). The cost of firm \(2\) is drawn by nature, but it is only observed by firm \(2\). The firms simultaneously choose their quantities.

   1. Calculate the Bayesian Nash equilibrium of the game.
   2. Calculate the market quantity and price for both states of the game.
   3. Calculate the firms' expected profits. That is, calculate the profits the firms expect before nature draws the marginal cost of firm \(2\). Which firm achieves greater profit? Why?
   1. Firm \(2\) observes its marginal cost and solves \[\max_{q_{s,2}} \left\{ p(q_{1} + q_{s,2})q_{s,2} - c_{s,2}q_{s,2}\right\} \quad\quad (c_{s,2} = 0,2).\] From this, we can calculate the two first order conditions (one for each state) as \[q_{s,2} = \frac{-c_{s,2} + 10 - q_{1}}{2} \quad\quad (c_{s,2} = 0,2).\] Further, we find that the expected quantity produced by firm \(2\) is \[\mathbb{E}q_{2} = \frac{9 - q_{1}}{2}.\] Firm \(1\) does not observe the marginal cost of firm \(2\), so it maximizes the expected profit \[\max_{q_{1}} \left\{ \mathbb{E} p(q_{1} + q_{2})q_{1} - q_{1}\right\},\] which gives the best response \[q_{1} = \frac{9 - \mathbb{E}q_{2}}{2}.\] Substituting the expected quantity produced by firm \(2\) gives \(q_{1} = 3\). Finally, substituting in the best response of firm \(2\), we get \[q_{0,2} = \frac{21}{6},\] when nature draws the low cost and \[q_{2,2} = \frac{15}{6},\] when nature draws the high cost.
   2. When the marginal cost of firm \(2\) is zero, the market quantity is \[q_{0} = q_{1} + q_{0,2} = 3 + \frac{21}{6} = \frac{39}{6},\] and the market price is \[p(q_{0}) = 10 - q_{0} = \frac{21}{6}.\] When the marginal cost of firm \(2\) is equal to \(2\), the market quantity is \[q_{2} = q_{1} + q_{2,2} = 3 + \frac{15}{6} = \frac{33}{6},\] and the market price is \[p(q_{2}) = 10 - q_{2} = \frac{27}{6}.\]
   3. The expected profit of firm \(2\) is \[\mathbb{E}\pi_{2} = \frac{1}{2} \frac{21}{6} \frac{21}{6} + \frac{1}{2} \left(\frac{27}{6} - 2\right)\frac{15}{6} = \frac{37}{4}.\] The profit of firm \(1\) is \[\mathbb{E}\pi_{1} = \frac{1}{2} \left(\frac{21}{6} - 1\right)3 + \frac{1}{2} \left(\frac{27}{6} - 1\right) 3 = 9.\] Firm \(2\) makes, on average, more profit than firm \(1\). Firm \(2\) has more information about the state of nature and can adjust its decisions better than firm \(1\).

1. Consider a Cournot duopoly with information asymmetry. The market inverse demand is \(p(q) = p_{0} + p_{1}q\). The cost function of firm \(1\) is \(c(q)=c_{1}q\), which is common knowledge. The cost of firm two is \(c(q)=c_{\theta,2}q\), where \(c_{\theta,2}\) is a positive random variable. The cost \(c_{\theta,2}\) is drawn by nature, but it is only observed by firm \(2\). The firms simultaneously choose their quantities.

   1. Calculate the Bayesian Nash equilibrium of the game.
   2. Calculate the market quantity and price.
   3. Calculate the firms' profits.
   4. Suppose that \(c_{1}>1\) and \(c_{\theta,2}=c_{1} + \theta\), where \(\theta=1\) with probability \(p\) and \(\theta=-1\) with probability \(1-p\). Which firm is producing more on expectation (Before nature draws the marginal cost of firm \(2\))? Why?
   1. Firm \(2\) observes its marginal cost and solves \[\max_{q_{\theta,2}} \left\{ p(q_{1} + q_{\theta,2})q_{\theta,2} - c_{\theta,2}q_{\theta,2}\right\}.\] From this, we can calculate the state dependent first order conditions (one for each state) as \[q_{\theta,2} = \frac{c_{\theta,2} - p_0 - p_{1}q_{1}}{2 p_{1}}.\] Further, we calculate the expected quantity produced by firm \(2\) as \[\mathbb{E}q_{2} = \frac{\mathbb{E}c_{2} - p_0 - p_{1}q_{1}}{2 p_{1}}.\] Firm \(1\) does not observe the marginal cost of firm \(2\), so it maximizes the expected profit \[\max_{q_{1}} \left\{ \mathbb{E} p(q_{1} + q_{2})q_{1} - c_{1}q_{1}\right\},\] which gives the best response \[q_{1} = \frac{c_{1} - p_0 - p_{1}\mathbb{E}q_{2}}{2 p_{1}}.\] Substituting the expected quantity produced by firm \(2\) gives \[q_{1} = \frac{2c_{1} - p_0 - \mathbb{E}c_{2}}{3 p_{1}}.\] Finally, substituting in the best response of firm \(2\), we get \[q_{\theta,2} = \frac{3 c_{\theta,2} + \mathbb{E} c_{2} - 2c_{1} - 2p_0}{6 p_{1}}.\]

   2. The total market quantity is

      \begin{align*} q_{\theta} &= q_{1} + q_{\theta,2} \\ &= \frac{2c_{1} - p_0 - \mathbb{E}c_{2}}{3 p_{1}} + \frac{3 c_{\theta,2} + \mathbb{E} c_{2} - 2c_{1} - 2p_0}{6 p_{1}} \\ &= \frac{2c_{1} + 3 c_{\theta,2} - \mathbb{E}c_{2} - 4p_0}{6 p_{1}}, \end{align*}

      and the market price is

      \begin{align*} p(q_{\theta}) &= p_{0} + p_{1}q_{\theta} = \frac{2c_{1} + 3 c_{\theta,2} - \mathbb{E}c_{2} + 2p_0}{6}. \end{align*}

   3. The profit of firm \(2\) is

      \begin{align*} \pi_{\theta,2} &= \left( \frac{2c_{1} + 3 c_{\theta,2} - \mathbb{E}c_{2} + 2p_0}{6} - c_{\theta,2} \right) \frac{3 c_{\theta,2} + \mathbb{E} c_{2} - 2c_{1} - 2p_0}{6 p_{1}} \\ &= - \frac{\left(2c_{1} - 3 c_{\theta,2} - \mathbb{E}c_{2} + 2p_0\right)^{2}}{36 p_{1}}. \end{align*}

      The profit of firm \(1\) is

      \begin{align*} \pi_{\theta,1} &= \left( \frac{2c_{1} + 3 c_{\theta,2} - \mathbb{E}c_{2} + 2p_0}{6} - c_{1} \right) \frac{2c_{1} - p_0 - \mathbb{E}c_{2}}{3 p_{1}} \\ &= \frac{-4c_{1} + 3 c_{\theta,2} - \mathbb{E}c_{2} + 2p_0}{6} \frac{2c_{1} - p_0 - \mathbb{E}c_{2}}{3 p_{1}} . \end{align*}
   4. We have \(q_{1} > \mathbb{E} q_{2}\) if and only if \[\frac{2c_{1} - p_0 - \mathbb{E}c_{2}}{3 p_{1}} > \frac{3 \mathbb{E} c_{2} + \mathbb{E} c_{2} - 2c_{1} - 2p_0}{6 p_{1}},\] which simplifies to \[c_{1} < \mathbb{E} c_{2}.\] By the distributional assumptions of \(c_{2}\), the last condition is equivalent to \[0 < \mathbb{E} \theta = 2p - 1.\] If the average shock to the marginal cost of firm \(2\) is positive (or equivalently \(p>1/2\)), then firm \(2\)'s production cost is on average greater than that of firm \(1\), and the expected production of firm \(2\) is less than that of firm \(1\).