Game theory attempts to mathematically. capture behavior in strategic situations, or. games, in which an individual s success in

Transcription

1 Game Theory

2 Game theory attempts to mathematically capture behavior in strategic situations, or games, in which an individual s success in making choices depends on the choices of others. A game Γ consists of a set of players, I, a set of moves/strategies, S available to those players, and a specification of payoffs for each combination of strategies, U. We say that Γ is finite, if S is. 1

3 Cooperative and Noncooperative Games A game is cooperative if the players are able to form binding commitments. For instance the legal system requires them to adhere to their promises. The solution concepts from cooperative game theory can be applied to arrive at revenue allocation schemes. In noncooperative games, binding commitments are not possible. 2

4 Zero-sum and Non-zero-sum Games In zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the equal expense of others). Examples: Poker, chess. In non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another. Most examples/situations in Economics correspond to this category 3

5 Simultaneous and Sequential Games Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players actions. Usually represented as normal-form Sequential (dynamic) games) are games where later players have some knowledge about earlier actions. This need not be perfect information about every action of earlier players. Usually represented as extensive form 4

6 Perfect and Complete information Games A game is one of perfect information if all players know the moves previously made by all other players. Thus, only sequential games can be games of perfect information. Ex: Ultimatum game Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions. 5

7 Normal (Strategic) form games We have a nonempty, finite set I of n N {1, 2,...} players The ith player, i I, has a nonempty set of strategies - her strategy space S i - from which she can choose one strategy s i S i. Assume that each player s strategy space is finite Example: Consider I = {1, 2}; S 1 = {Up, Down}; S 2 = {Left, Middle, Right} 6

8 s = (s 1,..., s n ) - the n-dimensional vector of individual strategies is called a strategy profile. For every different combination of individual choices of strategies we would get a different strategy profile s. The set of all such strategy profiles is called the space of strategy profiles S. It is simply the Cartesian product of the strategy spaces S i for each player 7

9 In the example above, where S 1 = {Up, Down}; S 2 = {Left, Middle, Right} S = n i=1 = S 1 S 2 = {(Up, Left), (Up, Middle), (Up, Right), (Down, Left), (Down, M iddle), (Down, Right)} 8

10 Player i is interested in what strategies the other n 1 players choose. We can represent such an (n 1)-tuple of strategies, known as a deleted strategy profile: s i = (s 1,...s i 1, s i+1,..., s n ) To each player i there corresponds a deleted strategy profile space S i, which is the space of all possible strategy choices s i by her opponents S i S 1 S 2... S i 1 S i+1... S n = jɛi\{i} S j 9

11 It is useful to write a strategy profile s S as a combination of her strategy s i S i and the deleted strategy profile s i S i of the strategies of her opponents. So, typically s (s i, s i ) In the example, S 1 = S 2 = {Left, Middle, Right}; S 2 = S 1 = {Up, Down} If s = (Up, Middle), then s 1 = Middle and s 2 = Up. 10

12 Payoffs The game is played by having all the players simultaneously pick their individual strategies. This set of choices results in some strategy profile S, which we call the outcome of the game. 11

13 Each player has a set of preferences over these outcomes. We assume that each player s preferences over lotteries over S can be represented by some von Neumann-Morgenstern utility function u i : S R. Therefore we can extend the domain of each u i to be the set of lotteries over outcomes in S. 12

14 We perform the extension such that u i has the expected-utility property. For any lottery over S of the form p.s + (1 p).s, where s, s S and p [0, 1], u i (p.s + (1 p).s ) = pu i (s) + (1 p)u i (s ) 13

15 At the conclusion of the game, then, each player i I receives a payoff u i (s) = u i (s i, s i ) The individual payoffs for all the n players for a particular strategy profile s define a payoff vector u(s) for that strategy profile u(s) (u 1 (s),..., u n (s)) i.e. u : S R n 14

16 Best responses to pure strategies We typically assume in game theory that all players are rational. This means that each player will choose an action which maximizes her expected utility given her beliefs about what actions the other players will choose. We ask the question: if player i knows which strategy each of her opponents will pick, what strategy should she pick? 15

17 Obviously, she should pick a best response to the plays of her opponents. Definition: We say that a strategy s i S i for player i is a best response by i to the deleted strategy profile s i S i iff ( s i S i ), u i (s i, s i ) u i(s i, s i ) 16

18 We will not necessarily have a best-response function which specifies player i s unique best response to some deleted strategy profile s i S i, but we will have a best-response correspondence for player i. BR i (s i ) = {s i S i : s i is a best response by i to s i } 17

19 Mixed strategies If every player plays a pure strategy, then the payoffs to all players are deterministic - there is no uncertainty concerning the payoffs. However, it is useful to expand each player s possible choices to include mixed strategies - randomizations over her pure strategies. 18

20 When a player i I chooses a mixed strategy, every other player j I \ i might be uncertain about which pure strategy s i S i the ith player will choose Formally, we say that a mixed strategy σ i for player i is a probability distribution over player is pure-strategy choices S i ; we write σ i (S i ) 19

21 We denote player is mixed strategy space by i. ( i = (S i )) A mixed strategy specifies a value on [0,1] for each s i S i. Each player chooses one and only one pure strategy s i S i in a single play of the game. Therefore any mixed strategy σ i (S i ) must be such that the sum of the probabilities associated with the pure strategies is unity. s i S i σ i (s i ) = 1 20