A Crash Course in Game Theory Werner Raub Workshop on Social Theory, Trust, Social Networks, and Social Capital II National Chengchi University – NCCU April 2011

Aim of the session A very brief introduction to game theory, introducing in an intuitive fashion the tools (concepts, assumptions, theorems) that we need for the subsequent session on “Game-theoretic models for effects of social embeddedness on trust and cooperation” and that are likewise theoretical background for many studies and applications that we discuss during this workshop

Social situations with interdependent actors and game theory as a tool for analyzing such situations Compare Max Weber’s famous definition of sociology and of social action: “Sociology […] is a science which attempts the interpretive understanding of social action in order thereby to arrive at a causal explanation of its course and effects […] Action is social in so far as […] it takes account of the behaviour of others and is thereby oriented in its course.” (Weber 1974: 88; emphasis added) Game theory: A set of tools (concepts, assumptions, theorems) for modeling and analyzing social situations with interdependent actors.

Game-theoretic models and the P-T-E scheme P – T - E Focus on “T” in P-T-E Focus on the two (!) ingredients of “T”: 1.Theory / theoretical model 2.Testable hypotheses that are generated from (ideally: deduced from) theory / theoretical model plus additional assumptions

Game-theoretic models and Coleman’s scheme Social conditions: interdependencies as summarized in the game description Individual preferences as summarized in the game description Individual strategy choices and equilibrium path behavior Collective effects: macro- outcomes such as Pareto- (sub)optimally Equilibrium behavior

Game theory: further reading – textbooks

Prisoner’s Dilemma R,RS,T T,SP,P CD D C Player 2 Player 1 Assumptions: T>R>P>S Simultaneous moves Binding agreements are not feasible (“noncooperative game”) Information: each player is informed on his or her own alternative actions and outcomes, as well as on alternative actions and outcomes for the partner

An intuitive characterization of 'goal-directed' action Actors have: Alternative actions Goals, i.e., they evaluate the possible outcomes of their actions Expectations (or information) on the “states of the world” (for example, expectations on certain “contingencies” or on the behavior of other actors). Assumption: actors choose the action that seems most appropriate, given their expectations, to realize their goals.

Goal-directed action in interdependent situations: some core concepts of game theory I Consider a noncooperative game between two players A and B (generalizing the definitions for the case of more than two players is straightforward). Let X be a strategy of actor A and let Y be a strategy of actor B. X is a best reply strategy of player A against strategy Y of player B if X maximizes A’s payoff against Y. –Note: Using strategy X against Y is consistent with goal-directed behavior of A, given that A anticipates B to use Y. X is a dominant strategy of player A if X is player A’s unique best reply against all strategies of player B. –Note: Goal-directed behavior implies that a player uses a dominant strategy. –Note: A player has at most one dominant strategy and often he or she has none.

Goal-directed action in interdependent situations: some core concepts of game theory II A strategy combination (X, Y) is a Nash equilibrium if X is a best reply strategy of player A against Y and Y is a best reply strategy of player B against X. –Note: Given that A anticipates B to use Y and B anticipates A to use X, playing Nash equilibrium is consistent with goal-directed behavior. –Note: Nash has shown that every “finite game” has at least one equilibrium, possibly in mixed strategies. –Note: A game often has more than one equilibrium.

John Nash

On John Nash: A Beautiful Mind

Goal-directed action in interdependent situations: some hypotheses of game theory H1: A player chooses a best reply strategy, given his or her anticipation of the strategy chosen by the other player H2: If a player has a dominant strategy, he or she will use this strategy H3: The chosen strategies will be a Nash equilibrium

Goal-directed action in interdependent situations: some core concepts of game theory III A strategy combination (X, Y) is Pareto-optimal if there is no other strategy combination that yields higher payoffs for at least 1 player, and not lower for the other player. A strategy combination (X, Y) is Pareto-suboptimal if there is another strategy combination that yields higher payoffs for at least 1 player, and not lower for the other player.

Application to the (non repeated) Prisoner’s Dilemma D is a dominant strategy. Hence, (D,D) is the unique Nash equilibrium. Hence, (D,D) will be played according to H2 as well as H3. (D,D) is Pareto-suboptimal. (C,C) is Pareto-optimal and better for both players than (D,D). But: (C,C) is not a Nash equilibrium!