Lecture Notes on Game Theory1 Tilman B¨orgers,Department of Economics, University of Michigan August 31, 2021 1. Definitions Definition 1. A strategic game is a list (N; (Ai)i2N ; (ui)i2N ) where N is a finite set of the form N = f1; 2; : : : ; ng (with n ≥ 2), for every i 2 NAi is a non-empty, finite set, and for every i 2 N ui is a real-valued function with domain A = ×i2N Ai. N is the set of players. Ai is the set of player i's actions, and ui is player i's von Neumann Morgenstern utility function. We write ai for elements of Ai. We refer to elements of Ai also as pure (as opposed to mixed) actions of player i. We write a for elements of A. We write A−i for ×j2N;j6=iAj. We write a−i for elements of A−i. Definition 2. A mixed action αi for player i is a probability distribution over Ai. We identify the mixed action αi that place probability 1 on action ai with this action itself. For any finite set X we write ∆(X) for the set of probability distributions over X, and thus ∆(Ai) is the set of mixed actions of player i. We write α for elements of ×i2N ∆(Ai). We write α−i for elements of ×j2N;j6=i∆(Aj). The expected utility of player i when players choose mixed action profile α is: 0 1 X Y (1) Ui(α) = @ui(a1; a2; : : : ; an) αj(aj)A (a1;a2;:::;an)2A j2N Note that this definition implicitly assumes that different players' random- izations when implementing their mixed action are independent. We shall make this assumption throughout. 2. Best Replies and Actions that are not Strictly Dominated Two basic ideas about rationality in games are (i) rational players only choose actions that maximize expected utility for some belief about the other players' actions, and (ii) rational players only choose actions that are not strictly dominated by other actions. Here, we formalize these two ideas, 1Definitions and notation in these notes are very similar, although not completely identical, to the definitions and notation in Osborne and Rubinstein [1]. 1 2 and then demonstrate that they imply exactly the same predictions about the behavior of rational players. Definition 3. A belief µi of player i is an element of ∆(A−i). An action ai 2 Ai is a best reply to a belief µi if X 0 (2) ai 2 arg max ui(ai; a−i)µi(a−i) : 0 a 2Ai i a−i2A−i Note that we do not require beliefs to be the products of their marginals, that is, we don't require player i to believe that the other players' actions are stochastically independent. If we did, Proposition 1 below would not be true. Definition 4. An action ai 2 Ai is strictly dominated by a mixed action αi 2 ∆(Ai) if (3) Ui(αi; a−i) > ui(ai; a−i) 8a−i 2 A−i: Note that we allow the strictly dominating action to be a mixed action. If we did not, then Proposition 1 below would not be true. ∗ Proposition 1. An action ai 2 Ai is a best reply to some belief µi 2 ∆(A−i) ∗ if and only if ai is not strictly dominated. ∗ Proof. Step 1: We prove the \only if" part, that is, we assume that ai ∗ is a best reply to a belief µi 2 ∆(A−i), and infer that ai is not strictly ∗ dominated. The proof is indirect. Suppose ai were strictly dominated by αi 2 ∆(Ai). Then, obviously, αi yields strictly higher expected utility given ∗ the belief µi than ai : X X ∗ (4) Ui(αi; a−i)µi(a−i) > ui(ai ; a−i)µi(a−i): a−i2A−i a−i2A−i ∗ We thus have that αi is a better response to µi than ai , which is almost what we want to obtain, but not quite. To obtain the desired contradiction, ∗ we want to find a pure action that is a better response to µi than ai . This can be done as follows. We re-write the left hand side of (4) as follows: X X X Ui(αi; a−i)µi(a−i) = αi(ai)ui(αi; a−i)µi(a−i) a−i2A−i a−i2A−i ai2Ai X X = αi(ai)ui(αi; a−i)µi(a−i) ai2Ai a−i2A−i 0 1 X X (5) = αi(ai) @ ui(ai; a−i)µi(a−i)A : ai2Ai a−i2A−i 3 Combining (4) and (5) we have: 0 1 X X X ∗ (6) αi(ai) @ ui(ai; a−i)µi(a−i)A > ui(ai ; a−i)µi(a−i): ai2Ai a−i2A−i a−i2A−i The left hand side of (6) is a convex combination of the expressions in large brackets in that term. This convex combination can be larger than the right hand side of (6) only if one of the expressions in large brackets is larger than the right hand side of (6), i.e., for some ai 2 Ai: X X ∗ (7) ui(ai; a−i)µi(a−i) > ui(ai ; a−i)µi(a−i): a−i2A−i a−i2A−i ∗ and thus ai is a better response to µi than ai , which contradicts the as- ∗ sumption ai is a best response to µi among all pure actions. 2 ∗ Step 2: We prove the \if" part, that is, we assume that ai is not strictly ∗ dominated, and we show that there is a belief µi 2 ∆(A−i) to which ai is a best response. The proof is constructive. We define two subsets, X and Y , of the set RjA−ij, that is, the Euclidean space with dimension equal to the number of elements of A−i. We shall think of the elements of these sets as payoff vectors. Each component indicates a payoff that player i receives when the other players choose some particular a−i 2 A−i. Now pick any one-to-one mapping f : A−i −! f1; 2;:::; jA−ijg. For any jA j action ai of player i, we write ui(ai; ha−ii) 2 R −i for the vector of payoffs that player i receives when playing ai, and when the other players play their various action combinations a−i. Specifically, the k-th entry of ui(ai; ha−ii) −1 −1 is the payoff ui(ai; f (k)) where f is the inverse of f. Intuitively, f defines an order in which we enumerate the elements of A−i, and ui(ai; ha−ii) lists the payoffs of player i when he plays ai and the other players play a−i in the order defined by f. The set X is: RjA−ij ∗ (8) X = fx 2 jx > ui(ai ; ha−ii)g: Here, \>" is to be interpreted as: \strictly greater in every component." Therefore, the set X is the set of payoff vectors that are strictly greater ∗ in every component than ui(ai ; ha−ii), that is the payoff vector that corre- ∗ sponds to the undominated action ai . The set Y is: jA−ij (9) Y = cofy 2 R j9ai 2 Ai : y = ui(ai; ha−ii)g: Here, \co" stands for \convex hull." The payoff vectors in Y are the payoff vectors that player i can achieve by choosing some mixed action. The weight that the convex combination that defines an element of y places on each 2An example and a graph that illustrate Step 2 follow after the end of the proof. 4 jA j element of fx 2 R −i j9ai 2 Ai : x = ui(ai; ha−ii)g corresponds to the probability which the mixed action assigns to each pure action ai 2 Ai. It is obvious that both sets are nonempty and convex sets. Moreover, their intersection is empty. If X and Y overlapped, then every common element would correspond to the payoffs arising from a mixed action of player i that ∗ strictly dominates ai . Because by assumption no such mixed action exists, X and Y cannot have any elements in common. In the previous paragraph we have checked all the assumptions of the separating hyperplane theorem: we have two nonempty and convex sets that have no elements in common. The separating hyperplane theorem (Theorem 1.68 in Sundaram [4]) then says that there exist some row vector π 2 RjA−ij which is not equal to zero in every component, and some δ 2 R, such that: (10) π · x ≥ δ 8x 2 X and (11) π · y ≤ δ 8y 2 Y: Here \·" stands for the scalar product of two vectors. We treat all vectors in X and Y as column vectors. Therefore, the above scalar products are well-defined. We now make two observations. The first is: ∗ (12) δ = π · ui(ai ; ha−ii): ∗ To show this note that by definition ui(ai ; ha−ii) 2 Y , and therefore, by (11), ∗ N ∗ n π · ui(ai ; ha−ii) ≤ δ. Next, for every n 2 define xn = ui(ai ; ha−ii) + " · ι, where " 2 (0; 1) is some constant and ι is the column vector in RjA−ij in which all entries are \1". Observe that for every n 2 N we have xn 2 X, so that by (10) we have: π · xn ≥ δ for every n. On the other hand, we ∗ have: limn!1 xn = ui(ai ; ha−ii). By the continuity of the scalar product ∗ of vectors therefore: π · ui(ai ; ha−ii) ≥ δ.