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MA300.2 Game Theory II, LSE MA300.2 Game Theory II, LSE Summary of Lecture 1 1. Games and Equilibrium Game in normal form: (S1,...,Sn; f1, . , fn). n is the number of players. Si is the strategy set of player i. Each fi is a real-valued payoff function defined on S ≡ S1 × ... × Sn. A pure strategy for player i is an element si ∈ Si.A pure strategy profile is an element s ∈ S; s = (s1, . , sn). A mixed strategy for i, µi, is a probability distribution over Si. µ = (µ1, . , µn) is a mixed strategy profile. Payoffs to i from mixed strategies are the obvious extension of fi: X fi(µ) ≡ fi(s)µ1(s1) · ... · µn(sn) s (use integrals if the strategy sets are not finite). Fix strategies µ−i of all the others and imagine maximizing i’s payoff: For given s−i, choose si ∈ Si to maximize fi(si, µ−i). If si solves above problem, call it a best response to µ−i. A mixed strategy can also be a best response: it must be a probability distribution over pure best responses. ∗ ∗ ∗ A profile µ is a Nash equilibrium if for every i, µi is a best response to µ−i. Theorem 1. [Nash] Every game with finite strategy sets for each player has a Nash equilib- rium, possibly in mixed strategies. Examples: Prisoner’s dilemma, coordination games, matching pennies. Pure and mixed strategy equilibria. Interpretation of mixed strategy equilibria as beliefs. 2. Efficiency Fundamental fact: one-shot games “typically” have inefficient outcomes. Of course, not always (recall the coordination game for instance or any zero-sum game for that matter). But there is a sense in which games have an intrinsic tendency towards inefficiency. Calculus the best way to see this. Example. Cournot oligopoly. n firms produce homogeneous output at constant marginal cost c ≥ 0. If a total of x is produced, then the market price is given by P (x) (this is a decreasing function). Joint monopoly output — call it m — the best outcome for the firms. To solve for m simply max[P (x) − c]x. 2 First-order condition: (1) P (m) + mP 0(m) − c = 0. To see if this can be “supported” as a Nash equilibrium assume that the other firms are doing their part, each producing m/n. Then a given firm will choose its output xi to max[P (xi + (n − 1)m/n) − c]x. First-order condition, this time for the best response xi: 0 (2) P (Q) + xiP (Q) − c = 0, where Q is total output. Try Q = m; then xi must be m/n. Substituting this into the left hand side of (2), we get the expression m (3) P (m) + P 0(m) − c. n Compare with (1). Notice that expression in (3) must be strictly positive. Understand where the externality lies: this is the source of the inefficiency. 3. Extensive Games 3.1. Strategies in the Extensive Form and the Basic Notion of Credibility. “Un- package” the Si’s. Stick to perfect information for the moment. Then an extensive game is a tree, in which each node is designated for a particular player, and the edges emanating from the node describe the actions available to her at that node. Warning: A strategy is now a function, a conditional list of actions. Example. Incumbent-Entrant game. Entrant decides to enter or not. Incumbent monopolist sees entry decision, then decides whether to fight the entrant or not. Payoffs are as follows: entrant gets 0 if she does not enter, -10 if she enters and incumbent fights, +10 if she enters and incumbent accommodates. Incumbent gets 20 if she’s alone, 10 if there is entry and she accommodates, and 5 if there is entry and she fights. Define strategies in this game. Define them again in the variant where incumbent does not observe the entry decision. Study the Nash equilibria of the original game. One of them is not “credible”. (The example has two goals, to get you to define strategies correctly, and to motivate the notion of “subgame perfection”.) 3.2. General Extensive Forms and Perfect Recall. More general extensive-form games allow for both sequential and simultaneous moves. Here are the main ingredients: 1. A set of players. 2. The order of moves (a tree). Three types of nodes: initial, non-initial decision node, terminal. Without loss of generality any “moves by Nature” can be placed at the inital node. 3. Payoffs assigned at every terminal node. 3 1 Initial node LR 2 Decision node ab c 1 Terminal Node dee d 4. What each player knows and chooses. Represented by information sets. Draw information sets to satisfy perfect recall: no two nodes are allowed to belong to the same information set if the player moving at the set should be able to distinguish between the two nodes based on her own past experience or behavior. A behavior strategy is a strategy which assigns probabilities over actions to a player at each information set at which it is her turn to move. This is the appropriate generalization of a mixed strategy when there is perfect recall. Theorem 2. [Kuhn] Under perfect recall, every mixed strategy (including mixtures of behav- ior strategies) is equivalent to some behavior strategy. 3.3. Subgame Perfection. A subgame is the continuation of a tree starting from a node which is a singleton information set. A subgame perfect equilibrium (SGPE) is a strategy profile which is a nash equilibrium, and furthermore indiuces a Nash equilibrium on every subgame. Reduces to the usual concept when there are no subgames, which is the case whenever all moves are simultaneous. Theorem 3. In any finite extensive game with perfect recall, a subgame perfect equilibrium always exists in behavior strategies. Proof. By induction on the number of nodes in a tree. Suppose that a SGPE exists for all trees with k nodes or less, for some k ≥ 2 (For k = 2, proof is trivial.) Consider an extensive form game with k + 1 nodes. If it has no proper subgames (apart from its terminal nodes) apply Nash (Theorem 1) to show existence of Nash equilibrium in mixed strategies. Then apply Kuhn’s Theorem (Theorem 2) to convert the mixed strategies to behavior strategies. If the game in question does have a subgame, fix payoffs from any SGPE in that subgame (must exist, by the induction hypothesis), and appen the payoffs to that node (i.e., artificially turn the node from which the subgame starts into a terminal node). Now we have one less node (at least) and on this artificial tree, a SGPE exists (again by induction). Expand this strategy profile by appending the equilibrium strategy profile from the subgame, and we’re done. 4 The proof of this theorem is closely related to the idea of backward induction, in which a decision tree is solved backwards from its terminal node. In game theory, backward induction (which we shall use liberally in this course) does raise some serious philosophical problems. Example. Rosenthal’s centipede game. Rosenthal’s Centipede 1212 1 22 1 100 100 2 1 4 3 98 97 100 99 0 3 2 5 96 99 98 101 The problem of rationality. How to update one’s belief at an information set which should not have been reached according to the logic of backward induction. Two fixes: (1) deviations as mistakes or “trembles”. (2) irrational types.
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