Lecture 9: The Agent Form of Bayesian Games

Transcription

1 Microeconomics I: Game Theory Lecture 9: The Agent Form of Bayesian Games (see Osborne, 2009, Sect 9.2.2) Dr. Michael Trost Department of Applied Microeconomics December 20, 2013 Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 1 / 23

2 Strategic form Nash equilibrium As stated in Theorem 8.4, the Bayesian equilibria of a Bayesian game are the Nash equilibria of its strategic form. Hence, a route of figuring out the set of Bayesian equilibria of a Bayesian game is to accomplish following steps: 1 transform the Bayesian game into its strategic form 2 find the Nash equilibria of this strategic game. Theorem 8.4 establishes that these Nash equilibria are the Bayesian equilibria of the Bayesian game. Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 2 / 23

3 Agent form of a Bayesian game In this lecture, we point to an alternative route of figuring out the Bayesian equilibria. This method is based on the transformation of a Bayesian game into its agent form. Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 3 / 23

4 Agent form of a Bayesian game The agent form of a Bayesian game is a strategic game in which the players are agents of the players of the Bayesian game. Each agent represents a player of the Bayesian game who is endowed with specific information about the actual state of the world. Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 4 / 23

5 Agent form Nash equilibrium Similar to Theorem 8.4, the Nash equilibria of the agent form prove to be the Bayesian equilibria of the Bayesian game, and vice versa. Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 5 / 23

6 Agent form Nash equilibrium This result opens an alternative route for finding the Bayesian equilibria. It runs as follows: 1 transform the Bayesian game into its agent form. 2 figure out all Nash equilibria of the agent form. According to the previous claim, these Nash equilibria represent all Bayesian equilibria of the Bayesian game. Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 6 / 23

7 Transformation of Bayesian games In order to transform the Bayesian game into its agent form, the following concepts are needed: signal function, agent of a player, posterior beliefs. Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 7 / 23

8 Signal function A signal function τ i of player i I is a function that specifies the information (the signal) the player receives in the actual state of the word. Formally, τ i : Ω Π i where τ i (ω) := P i (ω) holds for every state ω Ω. Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 8 / 23

9 Exercise: Bank run game EXERCISE: Determine the signal functions of our bank run game with asymmetric information. signal function of A τ A : signal function of B τ B : Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 9 / 23

10 Agent of a player An agent t i of a player i I is represented by the pair t i := (i, P i ) where i reveals the identity of the player behind this agent P i Π i is an information cell which specifies the information this agent of i has. The combination t i := (i, P i ) is also referred to as a type of player i. Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 10 / 23

11 Actual agent Consider some Bayesian game Γ and let ω Ω be some state of the world. In state ω, the player i receives the signal τ i and, thus, player i is of type (i, τ i (ω)). We term (i, τ i (ω)) the type (agent) of player i at state ω. Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 11 / 23

12 Exercise: Bank run game EXERCISE: Determine the set of agents of our bank run game with asymmetric information. set of agents of A T A := set of agents of B T B := Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 12 / 23

13 Posterior belief Let p i the prior of player i and P i Π i an information cell of player i I. The conditional probability measure p i (. P i ), which is defined by p i (E P i ) := p i(e P i ) p i (P i ) for every E Ω, is called the posterior belief of player i at information P i, or synonymously, the belief of the agent (i, P i ). Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 13 / 23

14 Belief of an agent Let t := (i, P i ) be an agent of Bayesian game Γ where p i denotes the prior of player i. Sometimes the belief of agent t is denoted simply by p t. That is, we define p t := p i (. P i ). Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 14 / 23

15 Exercise: Bank run game EXERCISE: Determine the beliefs of the agents of our bank run game with asymmetric information. belief of agent (A, {profit}) belief of agent (A, {loss}) belief of agent (B, {profit}) p A (. {profit}) : p A (. {loss}) : p B (. {profit, loss}) : Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 15 / 23

16 Agent form of a Bayesian game Definition 9.1 The agent form of a Bayesian game Γ := (I, Ω, (Π i ) i I, (p i ) i I, (A i ) i I, ( i ) i I ) is the strategic game Γ := (T, (A t) t T, ( t) t T ) consisting of a set T := i I T i of agents, for each agent t := (i, P i ) T, a set A t := A i of actions, for each agent t := (i, P i ) T, a preference relation t on A := t T A t which is representable by utility function U t(a) := ω Ω u i ( (a(i,τ i (ω))) i I, ω ) p i (ω P i ) where a := (a t ) t T A is some profile of agents actions, τ i the signal function of player i and u i the Bernoulli utility function of player i. Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 16 / 23

17 Agent form of a Bayesian game As stated in the previous definition, the agent form of Bayesian game is a strategic game with the following properties: the set of players is the set of agents. the action sets of the agents are the action sets of the Bayesian players behind these agents. the utility functions of the agents give the expected utilities resulting from the Bernoulli utilities of the Bayesian players behind these agents and from the beliefs of the agents. Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 17 / 23

18 Exercise: Bank run game EXERCISE: Determine the agent form of our bank run game with asymmetric information. set of agents action sets T := A t : preferences t: Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 18 / 23

19 Exercise: Bank run game EXERCISE: Depict the agent form of our bank run game with asymmetric information. withdraw retain withdraw retain withdraw retain withdraw retain Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 19 / 23

20 Characterization of Bayesian equilibrium Theorem 9.2 Let s := (s i ) i I be a strategy profile of Bayesian game Γ and a := (a t ) t T an action profile of the agent form game of Γ so that holds for every state ω Ω. a (i,p i (ω)) = s i (ω) Then, strategy profile s is a Bayesian equilibrium of the Bayesian game Γ if and only if action profile a is a Nash equilibrium of the agent form of Γ. Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 20 / 23

21 Agent form Nash equilibrium The previous theorem says that a Bayesian equilibrium of a Bayesian game is representable as a Nash equilibrium of its agent form. Due to this result the agent form of the Bayesian game can be used to figure out its Bayesian equilibria. As we will see, this method is often applied to solving Bayesian games. Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 21 / 23

22 Exercise: Bank run game EXERCISE: Determine the Nash equilibrium of the agent form of our bank run with asymmetric information. Agent (A, {profit}) Agent (A, {profit}) Agent (A, {loss}) withdraw retain withdraw (8.00,6.00,7.50) (8.00,4.00,8.50) retain (12.00,6.00,9.00) (12.00,4.00,10.00) Agent (B, {profit, loss}) withdraws Agent (A, {loss}) withdraw retain withdraw (10.00,10.00,10.00) (10.00,7.00,10.75) retain (12.00,10.00,10.00) (12.00,7.00,10.75) Agent (B, {profit, loss}) retains Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 22 / 23

23 Exercise: Bank run game EXERCISE: Deduce the Bayesian equilibrium of our bank run game from the Nash equilibrium of the agent form. The unique Bayesian equilibrium consists of the strategies sa : s A (profit) := s B : s B (profit) := sa (loss) := s B (loss) := Dr. Michael Trost Microeconomics I: Game Theory Lecture 9 23 / 23