Question

4. Consider a two-player game with strategy sets S₁ ={α₁,...,α_m} and S₂ = {β₁,...,β_n}.

(a) Suppose that (α_i,β_j) is a Nash equilibrium. Is it possible that a strategy α_j strictly dominates α_i?

(b) Assume again that (α_i,β_j) is a Nash equilibrium. Is it possible that another strategy α_j weakly dominates α_i?

Answer 1

We will formally define the games and solution concepts, and discuss the assumptions behind these solution concepts. In previous lectures we described a theory of decision-making under uncertainty. The second ingredient of the games is what each player knows. The knowledge is defined as an operator on the propositions satisfying the following properties: 1. if I know X, X must be true; 2. if I know X, I know that I know X; 3. if I don’t know X, I know that I don’t know X; 4. if I know something, I know all its logical implications. We say that X is common knowledge if everyone knows X, and everyone knows that everyone knows X, and everyone knows that everyone knows that everyone knows X, ad infinitum.

An n-player game is any list G = (S1,...,Sn; u1,...,un), where, for each i ∈ N = {1,...,n}, Si is the set of all strategies that are available to player i, and ui : S1 × ... × Sn → R is player i’s von Neumann-Morgenstern utility function. Notice that a player’s utility depends not only on his own strategy but also on the strategies played by other players. Moreover, ui is a von Neumann-Morgenstern utility function so that player i tries to maximize the expected value of ui (where the expected values are computed with respect to his own beliefs). We will say that player i is rational iff he tries to maximize the expected value of ui (given his beliefs).1 It is also assumed that it is common knowledge that the players are N = {1,...,n}, that the set of strategies available to each player i is Si, and that each i tries to maximize expected value of ui given his beliefs. When there are only 2 players, we can represent the (normal form) game by a bimatrix (i.e., by two matrices): 1\2 left right up 0,2 1,1 down 4,1 3,2 Here, Player 1 has strategies up and down, and 2 has the strategies left and right. In each box the first number is 1’s payoff and the second one is 2’s (e.g., u1 (up,left)=0, u2 (up,left)=2.)

For instance in this game, player 2 knows whether player 1 chose Head or Tail. And player 1 knows that when he plays Head or Tail, Player 2 will know what player 1 has played. (Games in which all information sets are singletons are called games of perfect information.) In this game, the set of strategies for player 1 is {Head, Tail}. A strategy of player 2 determines what to do depending on what player 1 does. So, his strategies are: HH = Head if 1 plays Head, and Head if 1 plays Tail; HT = Head if 1 plays Head, and Tail if 1 plays Tail; TH = Tail if 1 plays Head, and Head if 1 plays Tail; TT = Tail if 1 plays Head, and Tail if 1 plays Tail. What are the payoffs generated by each strategy pair? If player 1 plays Head and 2 plays HH, then the outcome is [1 chooses Head and 2 chooses Head] and thus the payoffs are (-1,1). If player 1 plays Head and 2 plays HT, the outcome is the same, hence the payoffs are (-1,1). If 1 plays Tail and 2 plays HT, then the outcome is [1 chooses Tail and 2 chooses Tail] and thus the payoffs are once again (-1,1). However, if 1 plays Tail and 2 plays HH, then the outcome is [1 chooses Tail and 2 chooses Head] and thus the payoffs are (1,-1). One can compute the payoffs for the other strategy pairs similarly. Therefore, the normal or the strategic form game corresponding to this game is HH HT TH TT Head -1,1 -1,1 1,-1 1,-1 Tail 1,-1 -1,1 1,-1 -1,1 Information sets are very important! To see this, consider the following game.

We have an object to be sold through an auction. There are two buyers. The value of the object for any buyer i is vi, which is known by the buyer i. Each buyer i submits a bid bi in a sealed envelope, simultaneously. Then, we open the envelopes, the agent i ∗ who submits the highest bid

bi∗ = max {b1, b2}

gets the object and pays the second highest bid (which is bj with j 6= i ∗). (If two or more buyers submit the highest bid, we select one of them by a coin toss.)