Solution Set 13 – Bayesian Games

1. (Exercise) Consider the case that two players have common prior on the space of states of nature, where each state represents the objective value of a commodity. Show that it is impossible that player 1 thinks after any signal he gets that the expected value is strictly above p*whereas player 2 thinks after any signal he gets that the expected value is strictly below p*. Explain why the common prior assumption is crucial for this result.

Let WÌÂ be a finite set of states, p a common prior over W, T_i signals of player i and t_i the signal functions with p(t_i)>0 for t_i ÎT_i, i=1,2. If player i thinks after any signal he gets that the expected value is strictly above p*,then å_{ti(w)= ti} wcond_i(w|t_i) >p* for any t_i ÎT_i. Then the expected value of w w.r.t p is: å_wÎW w p(w) = å _{ti ÎTi} p(t_i) å_{ti(w)= ti} w cond_i(w|t_i) >p*. On the other hand if player j thinks after any signal he gets that the expected value is strictly below p*, then the expected value of w w.r.t p is strictly below p*, a contradiction. The result does not hold without the common prior assumption because in that case, the two expectations of w above are computed w.r.t. different probability measures and need not be the same.

 

2 (Exercise) Construct and prove the one player analogous result of the proposition about the equivalence between the equilibria of the games G₁ and G₂ described above.

 

Suppose first that s₁ÎA₁^T1 is not a NE of the G₁ game. Then there is t₁ ÎT₁ and a₁ÎA₁ such that L₁(a₁,t₁)is strictly preferred by (1, t₁) to L₁(s₁(t₁),t₁), i.e.: å_{t1(w)= t1} cond₁(w|t₁) u₁(w, a₁ )> å_{t1(w)= t1} cond₁(w|t₁) u₁(w, s₁(t₁) ). Then defining s₁’ÎA₁^T1 by s₁’(t₁’)= s₁ (t₁’) for t₁’¹ t₁ and s₁’(t₁)= s₁ (t₁) we have: å_wÎW p₁ (w) u₁(w, s₁ ’(t₁(w)))= å _{t1’ ÎT1}  p₁ (t₁’)å_{t1(w)= t1’} cond₁(w|t₁’) u₁(w, s₁’(t₁’) ) > å _{t1’ ÎT1}  p₁ (t₁’) å_{t1(w)= t1’} cond₁(w|t₁’) u₁(w, s₁(t₁’) )=å_wÎW p₁ (w) u₁(w, s₁ (t₁(w))), since  p₁ (t₁)>0. So s₁’ is a profitable deviation from s₁  in the G₂ game, therefore s₁  is not a NE of the G₂ game.

 

Now let s₁ÎA₁^T1 be a NE of the G₁ game, then for any s₁’ ÎA₁^T1 and t₁ ÎT₁, L₁(s₁(t₁),t₁) is weakly preferred by player 1 to L₁(s₁’(t₁),t₁), i.e. å_{t1(w)= t1} cond₁(w|t₁) u₁(w, s₁(t₁) ) ³å_{t1(w)= t1} cond₁(w|t₁) u₁(w,s₁’(t₁)). Taking the expectation over all t₁ ÎT₁, we get: å_wÎW p₁ (w) u₁(w, s₁ (t₁(w)))= å _t1ÎT1  p₁ (t₁)å_{t1(w)= t1} cond₁(w|t₁) u₁(w, s₁(t₁) ) ³ å _t1ÎT1  p₁ (t₁) å_{t1(w)= t1} cond₁(w|t₁) u₁(w, s₁’(t₁) )=å_wÎW p₁ (w) u₁(w, s₁’ (t₁ (w))). So s₁  is also a NE of the G₂ game.

3. (More information may hurt)

Consider the Bayesian game in which N = {1, 2}, W = {w₁, w₂}, the set of actions of player 1 is {U, D}, the set of actions of player 2 is {L, M, R}, player 1's signal function is defined by t₁(w₁) = 1 and t₁(w₂) = 2, player 2's signal function is defined by t₂(w₁) = t₂(w₂) = 0, the belief of each player is (1/2, 1/2), and the preferences of each player are represented by the expected value of the payoff function defined as follows (where 0 < e < 1/2).

State w₁:

	L	M	R
U	1, 2e 1, 0 1, 3e 2, 2 0, 0 0, 3
D

State w₂:

	L	M	R
U	1, 2e 1, 3e 1, 0 2, 2 0, 3 0, 0
D

This game has a unique Nash equilibrium ((D,D), L) (that is, both types of player 1 choose D and player 2 chooses L). The expected payoffs at the equilibrium are (2, 2).

In the game in which player 2, as well as player 1, is informed of the state, the unique Nash equilibrium when the state is w₁ is (U, R); the unique Nash equilibrium when the state is w₂ is (U, M). In both cases the payoff is (1, 3e), so that player 2 is worse off than he is when he is ill-informed.

4. (Exchange game)

In the Bayesian game there are two players, say N = {1, 2}, the set of states is W = S × S, the set of actions of each player is {Exchange,Don't exchange}, the signal function of each player i is defined by t_i(s₁, s₂) = s_i, and each player's belief on W is that generated by two independent copies of F . Each player's preferences are represented by the payoff function u_i((X, Y), w) = w_j if X = Y = Exchange and u_i((X, Y), w) = w_i otherwise.

Let x be the smallest possible prize and let M_i be the highest type of player i that chooses Exchange. If M_i > x then it is optimal for type x of player j to choose Exchange. Thus if M_i ³ M_j and M_i > x then it is optimal for type M_i of player i to choose Don't exchange, since the expected value of the prizes of the types of player j that choose Exchange is less than M_i. Thus in any possible Nash equilibrium M_i = M_j = x: the only prizes that may be exchanged are the smallest.

5. (Second price auction)Consider a variant of the second-price sealed-bid auction described in the previous lecture in which each player i knows his own valuation v_i but is uncertain of the other players' valuations. Suppose that the set of possible valuations is the finite set V and each player believes that every other player's valuation is drawn independently from the same distribution over V. Players maximize their expected  “surplus”.

(a) Model this situation as the Bayesian game .

(b) Show that the game has a BNE where each player bids his valuation. Is it the unique BNE?

(c) Show that bidding is a weakly dominant action for each type of each player to bid his valuation.

 

Please see example 27.1 of Osborne and Rubinstein for part (a). Let us show part (c).  First let u_i(b|v_i) denote the payoff to player i when the bid profile is b=(b_i)_iÎNÎÂ₊^N  and the valuation profile is v=(v_i)_iÎNÎV^N , as in the second price auction game of exercise 2 in problem set 11. We had already shown in that context that it is a weakly dominant action to bid your own valuation, i.e., for any iÎN, b_-i =(b_j)_j¹iÎÂ₊^N-1  , v_i ÎV , and b_i ÎÂ₊: u_i(v_i , b_-i|v_i)³ u_i(b_i , b_-i|v_i). Now let a=(a(j, t_j))_{jÎN, tjÎTj} be a strategy profile of the associated G₁ game, iÎN, v_i =t_i ÎT_i , and and a_i ÎA_i= Â₊. Since u_i(t_i , (a(j, t_j))_j¹i |t_i)³ u_i(a_i , (a(j, t_j))_j¹i |t_i) for any t_-i=v_-iÎV^N-1 , we have that å_{ti(w)= ti} cond_i(w|t_i) u_i(t_i , (a(j, t_j(w)))_j¹i |t_i) ³ å_{ti(w)= ti} cond_i(w|t_i) u_i(a_i , (a(j, t_j(w)))_j¹i |t_i). Therefore it is weakly dominant for player (i, t_i) to bid his valuation t_i =v_i . Part (b) follows from part (c). This is not the unique BNE as before, for example the strategy profile where all types of player i always bid above the maximum possible valuation and all types of other players bid 0 is another BNE.