Incomplete information

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Scribe: Linji Yang

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Incomplete Information Game

The game is defined as: G=<N,\Omega,<A_i,u_i,T_i,\tau_i,p_i,C_i>_{i\in N}>, where

1. N is the set of players.

2. Ω is the set of the states of the nature. For instance, in a card game, it can be any order of the cards.

3. Ai is the set of actions for player i. Let A=A_1\times A_2\times ... A_N.

4. Ti is the types of player i, decided by the function \tau_i: \Omega \rightarrow T_i. So for each state of the nature, the game will have different types of players. For instance, in a car selling game, it will be the highest amount of money that player i is willing to pay for a specific car.

5. C_i \subseteq A_i \times T_i defines the available actions for player i of some type in Ti.

6. u_i: \Omega \times A \rightarrow R is the payoff function for player i. More formally, let L=\{(\omega,a_1,...,a_N)|\omega \in \Omega, \forall i, (a_i,\tau_i(\omega)) \in C_i\}, and u_i:L \rightarrow R.

7. pi is the probability distribution over Ω for each player i, that is to say, each player has different views of the probability distribution over the states of the nature. In the game, they never know the exact state of the nature.

The pure strategy s_i: T_i \rightarrow A_i should satisfy (s_i(t_i),t_i) \in C_i for all ti. So the strategy for each player only depends on his type, since he may not have any knowledge about other players' types. And the expected payoff to player i for such strategy profile is u_i(S)=E_{ \omega \sim p_i}[u_i( \omega ,s_1(\tau_1( \omega )),...,s_N(\tau_N( \omega )))]. Let Si be the set of pure strategies,

S_i = \{s_i: T_i \rightarrow A_i | (s_i(t_i),t_i) \in C_i for all ti}.

A Bayesian Equilibria of the game G is defined to be a (possibly mixed strategy) Nash equilibria of the game \hat{G}=<N,\hat{A}=S_1\times S_2 ... S_N, \hat{u} =u>. So for any finite game G, Bayesian Equilibria always exists.

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Example: The Car Auction Game

B NB
B 1, 1 2, 0
NB 0, 2 0, 0
nature = GOOD
B NB
B -1, -1 -2, 0
NB 0, -2 0, 0
nature = BAD

Let's first start with a simple version. The state of nature is GOOD/BAD, and player 1 believes that 60% of the time, the car is good, while player 2 believes that the chance of being good is 40%. Each player has two actions BID/NOTBID. Suppose there is no natural signal and hence no type of players, then after calculating the expected payoff table for each player, we can see that player 1 is going to BID, since it is a dominating strategy, and player 2 is not going to BID.

Now let's change the nature to Ω = (S,X,Y), where S is the condition of the car: GOOD/BAD, and X, Y are the set of opinions of each player: GOOD/BAD. The signal (τ1(ω),τ2(ω)) of each state ω = (s,x,y) is (x,y), that is after both players see the car, they have their own juedgements.

Further, we assign probability distrubtion pi over Ω as following: player 1 has a 60/40 prior on the cars being GOOD/BAD, while player 2 has a 40/60 prior on the cars being GOOD/BAD. Each player receives a signal. Both players believe that the process generating the signal is correct with probability a for player 1 and with probability b for player 2, independently. (In a more general case, they could have differing estimates, with ai,bi, but we take them to be the same for simplicity.) More formally,

p1(G,G,G) = 0.6ab
p1(G,G,B) = 0.6a(1 − b)
p1(G,B,G) = 0.6(1 − a)b
p1(G,B,B) = 0.6(1 − a)(1 − b)
p1(B,B,B) = 0.4ab
p1(B,B,G) = 0.4a(1 − b)
p1(B,G,B) = 0.4(1 − a)b
p1(B,G,G) = 0.4(1 − a)(1 − b)

and

p2(G,G,G) = 0.4ab
p2(G,G,B) = 0.4a(1 − b)
p2(G,B,G) = 0.4(1 − a)b
p2(G,B,B) = 0.4(1 − a)(1 − b)
p2(B,B,B) = 0.6ab
p2(B,B,G) = 0.6a(1 − b)
p2(B,G,B) = 0.6(1 − a)b
p2(B,G,G) = 0.6(1 − a)(1 − b)

In the homework, I will ask you to compute equilibria for a variety of values of a,b.

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