## Signaling Games

4.2.A Perfect Bayesian Equilibrium in Signaling Games A signaling game is a dynamic game of incomplete information involving two players a Sender (S) and a Receiver (R). The timing of the game is as follows 1. Nature draws a type f,- for the Sender from a set of feasible types T ii, , i according to a probability distribution p(ti), where p(i,) > 0 for every i and p(t ) +----1- p(ti) 1. 2. The Sender observes i, and then chooses a message rrij from a set of feasible messages M mi, , mj . 3....

## Final Offer Arbitration

Many public-sector workers are forbidden to strike instead, wage disputes are settled by binding arbitration. (Major league base ball may be a higher-profile example than the public sector but is substantially less important economically.) Many other disputes, including medical malpractice cases and claims by shareholders against their stockbrokers, also involve arbitration. The two major forms of arbitration are conventional and final-offer arbitration. In final-offer arbitration, the two...

## P nqi

Requirements 1 through 3 capture the spirit of a perfect Bayesian equilibrium. The crucial new feature of this equilibrium concept is due to Kreps and Wilson (1982) beliefs are elevated to the level of importance of strategies in the definition of equilibrium. Formally, an equilibrium no longer consists of just a strategy for each player but now also includes a belief for each player at each information set at which the player has the move. The advantage of making the players' beliefs explicit...

## LlB Iterated Elimination of Strictly Dominated Strategies

Having described one way to represent a game, we now take a first pass at describing how to solve a game-theoretic problem. We start with the Prisoners' Dilemma because it is easy to solve, using only the idea that a rational player will not play a strictly dominated strategy. In the Prisoners' Dilemma, if one suspect is going to play Fink, then the other would prefer to play Fink and so be in jail for six months rather than play Mum and so be in jail for nine months. Similarly, if one suspect...

## C A Double Auction

We next consider the case in which a buyer and a seller each have private information about their valuations, as in Chatterjee and Samuelson (1983). (In Hall and Lazear 1984 , the buyer is a firm and the seller is a worker. The firm knows the worker's marginal product and the worker knows his or her outside opportunity. See Problem 3.8.) We analyze a trading game called a double auction. The seller names an asking price, ps, and the buyer simultaneously names an offer price, pb. If pb > ps,...

## Existence of Nash Equilibrium

In this section we discuss several topics related to the existence of Nash equilibrium. First, we extend the definition of Nash equilibrium given in Section 1.1.C to allow for mixed strategies. Second, we apply this extended definition to Matching Pennies and the Battle of the Sexes. Third, we use a graphical argument to show that any two-player game in which each player has two pure strategies has a Nash equilibrium possibly involving mixed strategies . Finally, we state and discuss Nash's...

## Advanced Theory Mixed Strategies and Existence of Equilibrium

In Section 1.1.C we defined S, to be the set of strategies available to player i, and the combination of strategies s , ,s to be a Nash equilibrium if, for each player i, s is player i's best response to the strategies of the n 1 other players si, ,s _ ,s , s 1, ,sn gt Ui s , , s _i, Si, sf 1, , NE for every strategy s, in S,. By this definition, there is no Nash equilibrium in the following game, known as Matching Pennies. In this game, each player's strategy space is Heads, Tails . As a story...

## We

It remains only to check that the worker's strategy e L e with probability ir, e L e L with probability 1 7r e H e is a best response to the firms' strategy. For the low-ability worker, the optimal e lt e is e L and the optimal e gt eh is eh. For the high-ability worker, is superior to all alternatives. 4.2.C Corporate Investment and Capital Structure Consider an entrepreneur who has started a company but needs outside financing to undertake an attractive new...

## Static Games of Complete Information

In this chapter we consider games of the following simple form first the players simultaneously choose actions then the players receive payoffs that depend on the combination of actions just chosen. Within the class of such static or simultaneous-move games, we restrict attention to games of complete information. That is, each player's payoff function the function that determines the player's payoff from the combination of actions chosen by the players is common knowledge among all the players....

## Introduction to Perfect Bayesian Equilibrium

Consider the following dynamic game of complete but imperfect information. First, player 1 chooses among three actions L, M, and R. If player 1 chooses R then the game ends without a move by player 2. If player 1 chooses either L or M then player 2 learns that R was not chosen but not which of L or M was chosen and then chooses between two actions, L' and R', after which the game ends. Payoffs are given in the extensive form in Figure 4.1.1. Using the normal-form representation of this game...

## LlA Normal Form Representation of Games

In the normal-form representation of a game, each player simultaneously chooses a strategy, and the combination of strategies chosen by the players determines a payoff for each player. We illustrate the normal-form representation with a classic example The Prisoners' Dilemma. Two suspects are arrested and charged with a crime. The police lack sufficient evidence to convict the suspects, unless at least one confesses. The police hold the suspects in separate cells and explain the consequences...

## Dynamic Games of Complete but Imperfect Information

2.4.A Extensive-Form Representation of Games In Chapter 1 we analyzed static games by representing such games in normal form. We now analyze dynamic games by representing such games in extensive form.18 This expositional approach may make it seem that static games must be represented in normal form and dynamic games in extensive form, but this is not the case. Any game can be represented in either normal or extensive form, although for some games one of the two forms is more convenient to...

## Static Games of Incomplete Information

This chapter begins our study of games of incomplete information, also called Bayesian games. Recall that in a game of complete information the players' payoff functions are common knowledge. In a game of incomplete information, in contrast, at least one player is uncertain about another player's payoff function. One common example of a static game of incomplete information is a sealed-bid auction each bidder knows his or her own valuation for the good being sold but does not know any other...

## MPiP2MPIP2134

For every probability distribution p over Si, and p2 must satisfy for every probability distribution p2 over S2. Definition In the two-player normal-form game G S ,S2',ui,U2 , the mixed strategies p , p are a Nash equilibrium if each player's mixed strategy is a best response to the other player's mixed strategy 1.3.4 and 1.3.5 must hold. We next apply this definition to Matching Pennies and the Battle of the Sexes. To do so, we use the graphical representation of player i's best response to...

## Mixed Strategies Revisited

As we mentioned in Section 1.3.A, Harsanyi 1973 suggested that player j's mixed strategy represents player z's uncertainty about 's choice of a pure strategy, and that j's choice in turn depends on the realization of a small amount of private information. We can now give a more precise statement of this idea a mixed-strategy Nash equilibrium in a game of complete information can almost always be interpreted as a pure-strategy Bayesian Nash equilibrium in a closely related game with a little bit...

## Eo

9 If a consumer buys a good for price p when she would have been willing to pay the value v, then she enjoys a surplus of v p. Given the inverse demand curve Pi Qi a Q if the quantity sold on market i is Q the aggregate consumer surplus can be shown to be 1 2 Q . and assuming h lt a c tj, we have The results we derive are consistent with both of these assumptions. Both of the best-response functions 2.2.1 and 2.2.2 must hold for each i 1,2. Thus, we have four equations in the four unknowns h e...

## Problems

What is a static Bayesian game What is a pure strategy in such a game What is a pure-strategy Bayesian Nash equilibrium in such a game 3.2. Consider a Cournot duopoly operating in a market with inverse demand P Q a Q, where Q q q2 is the aggregate quantity on the market. Both firms have total costs cqi, but demand is uncertain it is high a an with probability 9 and low a ai with probability 1 0. Furthermore, information is asymmetric firm 1 knows whether demand is high or low, but firm 2...

## C

7Note that we have changed the notation slightly by writing , s,, s rather than Ui si,s2 . Both expressions represent the payoff to player i as a function of the strategies chosen by all the players. We will use these expressions and their -player analogs interchangeably. Solving this pair of equations yields which is indeed less than a c, as assumed. The intuition behind this equilibrium is simple. Each firm would of course like to be a monopolist in this market, in which case it would choose...

## Ri

Nash equilibria of this one-shot game correspond to subgame-perfect outcomes of the original repeated game. Let w,x , y, z denote an outcome of the repeated game zv, x in the first stage and y,z in the second. The Nash equilibrium li, l2 in Figure 2.3.4 corresponds to the subgame-perfect outcome li,l2 , li,l2 in the repeated game, because the anticipated second-stage outcome is Li, l2 following anything but Mi, m2 in the first stage. Likewise, the Nash equilibrium CRi, r2 in Figure 2.3.4...

## Normal Form Representation of Static Bayesian Games

Recall that the normal-form representation of an -player game of complete information is G Si S wi u , where S,- is player z's strategy space and u, si, ,s is player z's payoff when the players choose the strategies si, , s . As discussed in Section 2.3.B, however, in a simultaneous-move game of complete information a strategy for a player is simply an action, so we can write G Ai A U u , where A is player z's action space and Ui a , ,a is player z's payoff when the players choose the actions ,...

## Two Stage Games of Complete but Imperfect Information

We now enrich the class of games analyzed in the previous section. As in dynamic games of complete and perfect information, we continue to assume that play proceeds in a sequence of stages, with the moves in all previous stages observed before the next stage begins. Unlike in the games analyzed in the previous section, however, we now allow there to be simultaneous moves within each stage. As will be explained in Section 2.4, this simultaneity of moves within stages means that the games...

## Problems Section

In the following extensive-form games, derive the normal-form game and find all the pure-strategy Nash, subgame-perfect, and perfect Bayesian equilibria. 4.2. Show that there does not exist a pure-strategy perfect Bayesian equilibrium in the following extensive-form game. What is the mixed-strategy perfect Bayesian equilibrium 4.3. a. Specify a pooling perfect Bayesian equilibrium in which both Sender types play R in the following signaling game. b. The following three-type signaling game...

## Cournot Model of Duopoly

As noted in the previous section, Cournot 1838 anticipated Nash's definition of equilibrium by over a century but only in the context of a particular model of duopoly . Not surprisingly, Cournot's work is one of the classics of game theory it is also one of the cornerstones of the theory of industrial organization. We consider a very simple version of Cournot's model here, and return to variations on the model in each subsequent chapter. In this section we use the model to illustrate a the...

## ViU

Definition A pure-strategy perfect Bayesian equilibrium in a signal-ing game is a pair of strategies m ti and a mj and a belief fi ti my satisfying Signaling Requirements 1 , 2R , 2S , and 3 . If the Sender's strategy is pooling or separating then we call the equilibrium pooling or separating, respectively. We conclude this section by computing the pure-strategy perfect Bayesian equilibria in the two-type example in Figure 4.2.2. Note that each type is equally likely to be drawn by nature we...

## Dynamic Games of Complete and Perfect Information

The grenade game is a member of the following class of simple games of complete and perfect information 1. Player 1 chooses an action a from the feasible set Ai. 2. Player 2 observes d and then chooses an action a2 from the feasible set Ai- 3. Payoffs are U aj, a2 and 2 1, a2 . Many economic problems fit this description.2 Two examples 2Player 2's feasible set of actions, A2, could be allowed to depend on player l's action, a . Such dependence could be denoted by A a or could be incorporated...

## Dynamic Games of Complete Information

In this chapter we introduce dynamic games. We again restrict attention to games with complete information i.e., games in which the players' payoff functions are common knowledge see Chapter 3 for the introduction to games of incomplete information In Section 2.1 we analyze dynamic games that have not only complete but also perfect information, by which we mean that at each move in the game the player with the move knows the full history of the play of the game thus far. In Sections 2.2 through...

## Refinements of Perfect Bayesian Equilibrium

In Section 4.1 we defined a perfect Bayesian equilibrium to be strategies and beliefs satisfying Requirements 1 through 4, and we observed that in such an equilibrium no player's strategy can be strictly dominated beginning at any information set. We now consider two further requirements on beliefs off the equilibrium path , the first of which formalizes the following idea since perfect Bayesian equilibrium prevents player i from playing a strategy that is strictly dominated beginning at any...

## Bertrand Model of Duopoly

We next consider a different model of how two duopolists might interact, based on Bertrand's 1883 suggestion that firms actually choose prices, rather than quantities as in Cournot's model. It is important to note that Bertrand's model is a different game than Cournot's model the strategy spaces are different, the payoff functions are different, and as will become clear the behavior in the Nash equilibria of the two models is different. Some authors summarize these differences by referring to...

## Section

What is a game in normal form What is a strictly dominated strategy in a normal-form game What is a pure-strategy Nash equilibrium in a normal-form game 1.2. In the following normal-form game, what strategies survive iterated elimination of strictly dominated strategies What are the pure-strategy Nash equilibria 1.3. Players 1 and 2 are bargaining over how to split one dollar. Both players simultaneously name shares they would like to have, Si and s2, where 0 lt sj,s2 lt 1. If si 4- s2 lt...