Games with Uncertainty

Context

• The role of information is central in strategic interactions. Observing other players' actions before choosing leads to fundamentally different interactions and outcomes. Unobserved actions are not the only pieces of information relevant when players interact.

• In many social interactions, players are not fully aware of the state of nature of the game. They are unaware of the characteristics of other players with whom they interact.
  • How can we analyze interactions under such uncertain conditions?
  • How do economic agents interact with one another when information is incomplete?
  • Does uncertainty exacerbate or diminish the intensity of social dilemmas?

Course Structure Overview

Lecture Structure and Learning Objectives

Structure

• How to Win at Monopoly (Case Study)
• Bayesian Games
• A Bayesian Game Example
• Information Asymmetries
• Used Automobile Market Example
• Current Field Developments

Learning Objectives

• Illustrate the role of expectations in stochastic social interactions.
• Describe the types of uncertainty encountered in social interactions.
• Explain how the concept of equilibrium is adjusted to account for uncertainty.
• Describe how the asymmetry of information can lead to economic inefficiencies.

How to Win at Monopoly

• Monopoly is one of the most popular board games.

• Each player’s objective is to force her friends and family into poverty.
• It ends when all but one player go bankrupt. Or when someone flips the board in the midst of cheating accusations.

Do Strategies Matter?

What Should we Expect?

The Best Bang for your Buck

Take-Home Strategies

• 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.

Information in Game Theory

• Information uncertainty can be classified into two broad categories.
  • Uncertainty concerning the actions of others.
  • Uncertainty concerning the types of others.

• Both categories of uncertainty can coexist in social interactions.

Imperfect Information

• 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.

Incomplete Information

• 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.

Bayesian Games

• A game with incomplete information, in which the probability distribution of types is common knowledge to all players, and players reach their decisions by calculating Bayesian probabilities, is called a Bayesian game.
• A Bayesian game is a description of a social interaction specifying
  • the players (who is participating?),
  • the feasible actions (when is someone playing? What can she do?)
  • the information (what is known by players when making their decisions?)
  • the payoffs (what is the outcome for each possible combination of actions?)
  • the states of nature (what can be the types of the players in the game?)
  • the probability distribution of states (how probable is each type combination?)

A Bayesian Game Example

• Consider a random variable \(x\) such that
  • \(x=12\) with probability \(2/3\) and
  • \(x=0\) with probability \(1/3\)
• Player \(A\) observes \(x\) before playing, while player \(B\) does not.

Bayesian Extensive Form

• 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.

A Bayesian Extensive Form Example

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

Bayesian Normal Form

• 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.

An Example of Strategies in Bayesian Normal Form

• 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 \(\left\{Left, Right\right\}\).

An Example of Payoffs in Bayesian Normal Form

• 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*}

A Bayesian Normal Form Example

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

Bayesian Nash Equilibrium

• A collection of strategies, one for each player and for each state, such that each strategy constitutes a best response to the remaining players’ strategies, is called a Bayesian Nash equilibrium (typically abbreviated BNE).
• For static Bayesian games, the Bayesian Nash equilibria can be obtained by calculating the Nash equilibria of the Bayesian normal form

A Bayesian Nash Equilibrium Example

• What is the Bayesian Nash equilibrium of the previous Bayesian game example?
• \(BNE = \left\{ \left\{Bottom_{12}Bottom_{0}, Right\right\} \right\}\)

Information Asymmetry

• In games with incomplete or imperfect information, there can be imbalances between the information available to the players.
• In stochastic games, we say that there is information asymmetry whenever at least one of the agents has access to information unavailable to other agents.
• Information asymmetries can crucially affect the balance of power between economic agents.

Used Automobile Market Game

• 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 (\(q\), \(1-q\)).

When does Trade Take Place?

• 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.

Peaches and Lemons

The Automobile Market

• The Bayesian normal form of the game is

Equilibria without Trade

• 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 \(\left\{TT, N\right\}\).

Equilibria with only Lemons

• 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 \(\left\{NT, T\right\}\).

Equilibria with Lemons and Peaches

• 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 \(\left\{TT, T\right\}\).

Current Field Developments

• Most models used in market calibrations and estimations are stochastic.
• Dynamics and stochasticity are commonly both present in economic models.
• Future is usually modeled as being uncertain.
• Finding equilibria in dynamic, stochastic games continues to be an open problem.
• Systematic ways to find all or select particular equilibria in such games are unknown.
• Recent work in game theory also includes information processing cost in the analysis.

Concise Summary

• There are two basic types of information uncertainty in social interactions.
  • Imperfect information (some actions of other agents are not observed).
  • Incomplete information (some types of other agents are not observed).
• The Bayesian Nash equilibrium extends the idea of Nash equilibrium in stochastic games.
• Information asymmetries affect the balance of power in stochastic games.
• Information asymmetries can lead to economic inefficiencies.

Further Reading

• Belleflamme and Peitz (2010, sec. A.3)
• Watson (2008, chaps. 24, 26, 27)

Exercises

Group A

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.

Group B

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.

References

References

Topic's Concepts

• first price sealed bid auction
• second price sealed bid auction
• information asymmetry
• bayesian nash equilibrium
• bayesian normal form
• bayesian extensive form
• harsanyis transformation
• bayesian game
• incomplete information game
• imperfect information game