Elements of Economic Analysis II Lecture X: Introduction to Game Theory Kai Hao Yang 11/14/2017 1 Introduction and Basic Definition of Game So far we have been studying environments where the economic agents decision do not affect each others directly. In the studies of competitive markets, all the consumers and firms are price takers and the optimal consumption/production affects each other only through equilibrium prices. In the studies of a monopoly, only the monopolist is making decisions. In game theory, however, each economic agent s decision affects the others directly. That is, game theory is a study of strategic interactions in which when making decisions, each economic agents will take the others behavior (and thus preference) into account. We begin with a simple example. Suppose that there are two suspects are investigated by a prosecutor. The prosecutor does not have direct evidence to convict them for a serious crime so she is hoping that the suspects will confess themselves. If both of them do not confess, the prosecutor can only convict them with a less serious crime. On the other hand, if both of them confess, both will be convicted. However, if only one of them confesses, the confessor will become a state witness and the other suspect will be sentenced even more seriously. For simplicity, suppose that when both do not confess, both will get a payoff of 3. If both confess, both will get a payoff of 2. If only one of them confesses, the confessor will get a payoff of 4 and the other Department of Economics, University of Chicago; e-mail: khyang@uchicago.edu 1
2 will get a payoff of 1. The payoffs the result from their decisions can be summarized by the following table. Table 1: Prisoner s Dilemma Confess Not Confess Confess (3,3) (1,4) Not Confess (4,1) (2,2) Notice that although both confessing would be the best outcome for them jointly. For each of the suspects, not confessing is the best thing to do irrespective of the other s decision. As a result, the unique prediction of this game is that both will not confess. We now formally define a game. A strategic form game is defined by three elements. Players, Strategies and Payoffs. Formally, N is the set players. For each i N, S i is player i s strategy space. Also, for each i N, u i : S i R i N is player i s payoff function. Notice that the payoff function u i has domain i N S i, which means that each player s payoff depends on the others strategies directly. Together, we say that a strategic form game is defined by (N, (S i ) i N, (u i ) i N ). For example, the example above is a strategic form game. The set of players is {1, 2}. The strategy spaces are S i = {Confess, Not Confess} and payoff functions are given by Table 1. We say that a game if finite if N is a finite set and S i is finite for each i N. 2 Pure Strategy Nash Equilibrium With the formal definition of a game, we can then begin analyzing it. The first step is to define a proper solution concept of a game. There are many well-defined solution concepts in game theory. We start with the most common and widely used one Nash equilibrium. Briefly, Nash equilibrium is a strategy profile (a description of all the players strategy) such that no one will have incentive to deviate unilaterally. That is, a Nash equilibrium is a collective behavior that, given that all the other players in the game are not moving, each
3 player will no incentive to change their strategy. Formally, given a strategic form game (N, (S i ) i N, (u i ) i N ), a strategy profile s = (s i ) i N i N S i is a Nash equilibrium if for all i N, for all s i S i, u i (s) u i (s is i ), where s i = (s j ) j i denotes the strategy profile of all the players except i under s. For example, in the Prisoner s Dilemma above, (Confess, Confess) is the unique Nash equilibrium, since this is the only strategy profile in which each player will not have incentive to deviate given that the other player is choosing to Confess. As another example, consider the following game, sometimes referred as Battle of Sexes. Imagine a situation where a couple, Bob and Alice, has to decide whether they should go to a baseball game or to a movie tonight. Suppose that Bob really wants to see the movie and Alice really wants to watch the game. However, it will be the worst for both of them if they cannot go to either of them together. Suppose that they are separated in the morning because they work in different places and their phones are broken so that they cannot communicate. As such, they cannot see each other s decision when making their owns. We summarize Bob s and Alice s payoff in Table 2. To solve for a Nash equilibrium, the following procedure will be useful. From Bob s perspective, given that Alice chooses to go to the game, his best strategy is to go to the game too (so we put an asterisk on Bob s payoff here); Given that Alice chooses to go to the movie, his best strategy is to go to the movie too. On the other hand, from Alice s perspective, given that Bob goes to the movie, her best strategy is to go to the movie too; Given that Bob goes to the game, her best strategy is to go to the game too. After this procedure, we have identified what are the best strategies as a response to the other s strategies. It is then sufficient to find the scenarios that they are consistent with each other: Bob s strategy is the best given Alice s strategy and Alice s strategy is also the best given Bob s. This is exactly the strategy profiles that gives two asterisks in Table 2, which are exactly the Nash equilibria of this game. The Battle of Sexes gives an example for a game that has more than one Nash equilibria. It is also possible that a game may not have any Nash equilibrium at all. Consider the following example, called Matching Pennies, in which player 1 and 2 are flipping coins and 1 always wants to have the same side as 2 does and 2 always does not. You can verify that
4 Table 2: Battle of Sexes Game Movie Game (2*,1*) (0,0) Movie (0,0) (1*,2*) there are no Nash equilibria in this game, since for any strategy profile, either player 1 or player 2 would want to deviate to the other side. Table 3: Matching Pennies Head Tail Head (1*,0) (0,1*) Tail (0,1*) (1*,0) 3 Mixed Strategy Nash Equilibrium and Best Response Correspondence The previous example indicates an undesirable feature of the Nash equilibrium defined above. After all, for a solution concept to have predictive/explanatory power, it had better be the case that a solution always exists. To address this issue, we can extend the definition of Nash equilibrium and allow randomizations. That is, we can allow the players to randomize among strategies. Formally, let (N, (S i ) i N, (u i ) i N ) be finite a strategic form game. For each i, let M i = {σ i [0, 1] Si σ i (s i ) = 1} s i S i be the set of probability distributions on player i s strategy space and let M = i N M i. Also, for each i let v i : M R be defined as: v i (σ) = s S u i (s) i N σ i (s i )
5 denote the expected payoff of buyer i given that each buyer j N randomizes according to σ j. Together, (N, (M i ) i N, (v i ) i N ) is another strategic form game, which we call the mixed extension of the original game. With a mixed extension, Nash equilibrium can be defined analogously. Specifically, a strategy profile σ = (σ i ) i N M is a Nash equilibrium if for all i, for all σ i M i, v i (σ) v i (σ i, σ i ). We referred this extended Nash equilibrium as mixed strategy equilibrium. At a glance, it seems a lot harder to solve for a mixed strategy equilibrium comparing to what we have done before. However, the following techniques are going to be useful for finding mixed strategy equilibria. First, observe that for each player i, given the opponent s strategy σ i, player i will optimally choose some σ i argmax v i (σ i, σ i ). σ i M i If we let β i (σ i ) be defined as argmax σ i M i v i (σ i, σ i ), then β( ) is a set-valued function (or, a correspondence), that gives the set of optimal choices for player i given that the other players are playing σ i. This is called the best response correspondence for player i. Finding mixed strategy Nash equilibrium is then reduced to finding some strategy profile σ = (σ) i N such that σ (β i (σ i )) i N, which is essentially solving a system of equations. For example, consider the Matching Pennies game defined above. If we allowed players to randomize, each player s strategy is then a probability p i [0, 1] of choosing Head. Figure 1 graphs the best response correspondences for player 1 and 2 together. As we can see in Figure 1, there is an intersection between the two best response correspondences, which gives p 1 = p 2 = 1/2. Therefore, each player randomizing by probabilities 1/2 and 1/2 is the unique mixed strategy Nash equilibrium in this game.
6 p 2 β 2 (p 1 ) 1 2 β 1 (p 2 ) 0 1 2 Figure 1. p 1 In fact, with two-players and two-strategies games, the procedure can be further simplified. Observe that for each i, if σ i β i (σ i ), then for all s i such that σ i (s i ) > 0, v i (s i, σ i ) must be the same. Since if not, if v i (s i, σ i ) < v i (s i, σ i ) for some s i, s i such that σ i (s i ) > 0 and σ i (s i) > 0, i can choose a strategy that removes all the probability weights on s i to s i, which will then strictly improve payoff and therefore σ i can not be in β i (σ i ). Using this observation, we know that in a game with two strategies, in any mixed strategy equilibrium, players must be indifferent between choosing two strategies given the other player s strategy. If there are only two payers, the problem is then simplified to be solving a system of linear equations with two unknowns. For example, in the Matching Pennies game, let p i be the probability that i is choosing Head. For player 1, player 2 s strategy must be such that he is indifferent. That is: p }{{} 2 = (1 p 2 ) }{{}. Expected payoff when choosing Head Expected payoff when choosing Tail Similarly, for player 2 to be indifferent, it must be that p 1 = (1 p 1 ). Combining both of them gives p 1 = p 2 = 1/2.
7 For another example, consider the Battle of Sexes game, for 1 to be indifferent, it has to be that 2p 2 = (1 p 2 ) and for 2 to be indifferent, it has to be that p 1 = 2(1 p 1 ) Solving both equations give p 1 = 2 3, and p 2 = 1 3. As such, in addition to the two pure strategy equilibria (Game, Game) and (Movie, Movie), player 1 randomizes with probability 2/3 and 1/3 while player 2 randomizes with probability 1/3 and 2/3 is also an equilibrium.