Game theory and applications: Lecture 1 Adam Szeidl September 20, 2018 Outline for today 1 Some applications of game theory 2 Games in strategic form 3 Dominance 4 Nash equilibrium 1 / 8
1. Some applications of game theory 1 Political economy. Citizens and the elite interact to determine democratic vs non-democratic political institutions. 2 Industrial organization. Firms profits depend on their prices and products, but also on the prices and products offered by other firms. 3 Market design. How should we organize markets, such as auctions or kidney exchange, to take into account strategic interaction and maximize gains from trade. 4 Finance. Whether to withdraw money from a bank, or to short a particular asset or currency, depends on how central banks and other traders in the economy act. 5 International trade. Interest groups and voters compete to influence politicians setting tariff levels. 6 Labor economics. Firms try to design incentive schemes and structure compensation to affect the behavior of their workers. 2 / 8
2. Games in strategic form Players i {1, 2,..., I }. Pure strategy spaces S i are assumed to be finite except if noted otherwise. Finiteness makes math easier when proving theorems, but rules out standard econ examples like auctions where we like to use calculus. S = i S i is the space of strategy profiles s = (s 1,..., s I ). Von Neumann-Morgenstern utility functions u i : S R Payoff of a lottery over strategies is the expected value. Utility represents preferences, not monetary payoff. 3 / 8
Pure and mixed strategies For now think of strategies as buttons and imagine players sit in different rooms and cannot talk. A mixed strategy σ i Σ i is a probability distribution over S i. Σ i is prob simplex of non-negative vectors in R #S i that sum to 1. Players randomize independently, so that Pr(s 1 = s 1, s 2 = s 2 ) is σ 1 (s 1 ) σ 2(s 2 ). Note that S i Σ i and S Σ. We denote by s i strategy of players not i, so that (s i, s i) = (s 1,..s i 1, s i, s i+1,..., s I ) i s payoff to a mixed strategy profile σ is her expected utility u i (σ) = ( I ) σ j (s j ) u i (s). s S j=1 Linear in own probabilities σ i. Polynomial in σ i and hence continuous. 4 / 8
3. Dominance Definition. s i is strictly dominated if there exists σ i such that u i (σ i, σ i) > u i (s i, σ i ) for all σ i. A rational player (preferences represented by expected utility) who knows the structure of the game should not use these. Same definition applies to mixed strategies σ i. 1 Equivalent to u i (σ i, s i) > u i (s i, s i ) for all s i. 2 A pure strategy can be dominated by a mixed strategy even when it is not dominated by any pure strategy. 3 A best response is not strictly dominated. Def.: σ i is a best response to σ i if it is an element of arg max σ i u i (σ i, σ i) In 2-player games the converse is also true (follows from separating hyperplane theorem, FT2). 4 A mixed strategy putting positive prob on a strictly dominated pure strategy is strictly dominated. The converse is false. 5 / 8
Iterated strict dominance (ISD) Delete strictly dominated strategies, this makes more strategies dominated, repeat, etc. Each step in iteration requires another round of i knows that j knows that... that everyone is rational Formal definition complicated because of mixed strategies (FT 2). Ends in finite number of rounds in a finite game. Fact: outcome set does not depend on order of deletion. Intuitively, cannot make a strictly dominated strategy undominated by deleting dominated rows/columns. If process yields a single strategy profile, game is said to be solvable by iterated strict dominance. Not always a great prediction as ISD supposes common knowledge of rationality. ISD is particularly questionable in games with extreme payoffs. 6 / 8
Application: games vs decisions Key difference: In decision problems agent cannot be worse off by having more choices or higher payoffs to some outcomes, but in games this needn t be true. Formally, in decision problems: B A = max a B u (a) max u (a), and a A a A, u (a) u (a) = max a A u (a) max a A u (a) Neither of these hold when looking at utilities resulting from ISD applied to a game. Intuitively, decision theory applies when holding fixed opponent actions, but in games these change when environment and my action change. Similar pattern holds with information: knowing more can make agent worse off in a game but not in a decision problem. 7 / 8
4. Nash equilibrium Definition. Strategy profile σ is a Nash equilibrium (NE) if for all players i u i ( σ i, σ i) ui ( σi, σ i) for all σi Σ i. If in a NE i assigns positive probability to multiple pure strategies, all of them must yield same payoff. s is a strict NE if u i ( s i, s i) > ui ( si, s i) for all si S i. Only pure strategy equilibria can be strict. Strict equilibria need not exist. Fact: if a game is solvable by ISD, solution is the unique NE of the game. And ISD never eliminates a NE. 8 / 8