Rational Choice Sociology Lecture 5 Game theory I: the concept and classification of games

The Concept of Game I Game theory is the branch of RCT that analyzes the problem of rational choice in strategic situation; or: theory of strategically rational action Strategic situation is the situation of choice involving at least two rational actors making interdependent choices and knowing about this interdependence Interdependence means that outcomes of alternative choices of each actor depend on what other actors choose Game is analytical model of a strategic situation

The Concept of Game II Actor2 C1 C2 r11 r12 r21 r22 r11 r12 r21 r22 (Columns) Actor 1 (Rows) C1 C2 Action 1 r11 r12 Action 2 r21 r22 Action1 Action2 r11 r12 r21 r22

The Concept of Game III The parametric situations can be considered as involving the game against nature (including people behaving like nature) Strategic actions can be considered as the subclass of the social actions in the sense of Max Weber. Social action: action oriented to the actions of other actors. Strategic action is instrumentally rational social action in the strategic situation The social actions of at least two actors constitute a game (or can be described as game) if following conditions are satisfied: Each actor is instrumentally rational (maximizes utility or expected utility): this means that her preferences are consistent (satisfy axioms discussed above) Each actor knows (1): assumption of symmetry Each actor knows that other actors know that she knows (2): = assumption of the common knowledge of rationality From the same information each actor makes the same conclusions (no misperceptions or misunderstandings): = assumption of common alignment of beliefs

The Concept of Game IV The description of a game includes the specification of: Actor set N {1,2,.., n}, including at least two players Strategy set S {1,2, …, m}, including at least two strategies Outcome set O. The size of this set depends on how many players there are and how many strategy choices each of them has. E.g. if there are two players and each of them has 2 strategies, there are 2×2= 4 outcomes; if there are two players, one of them has 2, another 3 strategies, there are 2×3=6 outcomes; if there are 3 players, one of them has 2, another 4, and the third one 6 strategies, there are 2×4×6=48 outcomes etc. Names of the outcomes can be built using as elements the names of strategies that cause them Utility functions U that for each player and for each outcome ascribe the utility index u (ordinal or cardinal)

The Concept of Game V s1 s2 3 s1s1 3 1 s1s2 4 4 s2s1 1 2 s2s2 2 Actor 2 Actor 1 Actor 1: s2s1 > s1s1 > s2s2 >s1 s2 4 3 2 1 Actor 2: s1s2 > s1s1 > s2s2 > s2s1 4 3 2 1 s1 s2 3 s1s1 3 1 s1s2 4 4 s2s1 1 2 s2s2 2

The Concept of Game VI s1 s2 Actor 1 Actor 1: s2s1 > s1s1 > s2s2 >s1 s2 22(4) 20(3) 14(2) 3(1) Actor 2: s1s2 > s1s1 > s2s2 > s2s1 -2(4) -4(3) -30(2) -55(1) s1 s2 20 s1s1 -4 3 s1s2 -2 22 s2s1 -55 14 s2s2 -30

The Concept of Game: an Example (Prisoner’s Dilemma) Bill John The police have caught two suspected drug dealers, John and Bill. They are now sitting in two separate cells in the police station and cannot communicate. The prosecutor tells them that they have one hour to decide confess their crimes or deny the charges. The prosecutor has enough evidence to prove the charge to sentence them for 2 years for a well-documented offences. However, to sentence them for drug trafficking the prosecutor should have the confession of at least one of them. The legal situation is as follows: if both prisoners confess they will get 8 years each. However, if one confesses and the other does not, then the prisoner who confesses will be rewarded and get away with just 1 year in prison, whereas the other will get 10 years. Do not confess (cooperate) Confess (defect) -2 s1s1 -2 -10 s1s2 -1 -1 s2s1 -10 -8 s2s2 -8

Taxonomy (classification) of games Two-person and N-person (N>2) (How many players) Cooperative and non-cooperative games (in cooperative games players can agree on binding contracts that force them to respect whatever they have agreed) Non-iterated (one-shot) and iterated games Simultaneous-move versus sequential-move games Games with perfect information versus games with imperfect information (sequential game is a game with perfect information if players have full information about the strategies played by the other players in earlier rounds; if they have only some information about their previous moves, the sequential game is with imperfect information) Constant (including zero) sum games versus non-constant sum

Two-Person Constant-Sum Games A game is constant-sum if for all outcomes the sum of payoffs for all players is constant. A special case of constant-sum game is zero-sum game where this sum is zero. All constant-sum games sometimes are called zero-sum, because each constant-sum game can be transformed into a zero-sum game by subtracting from each payoff half of the sum of the payoffs

Two-person zero-sum game 2 3 4 1 5 0 1 4 2,5 2,5 0 5 s1 s2 s3 -0,5XX 0,5ZZ 1,5 -1,5Z 2,5 -2,5Z -0,5 0,5Z -1, 5X 1,5 0 0 -2,5X 2,5 1,5 -1,5Z

Two-Person Zero-Sum Games Zero-sum games are models of so-called antagonistic conflicts (e.g. war). For each outcome, one player can gain only as much the other player loose. Zero-sum games is the oldest province of the game theory. They were investigated in an exhaustive way in the book J. von Neumann and O. Morgenstern Theory of Games and Economic Behavior (1944) In this book, the authors proved that in the zero-sum games the choice according to minimax(maximin) rule is rational (utility-maximizing) for all players A pair of strategies are in equilibrium if the outcome determined by the strategies equals the minimal value of the row and the maximal value of the column (minimax condition rule)

Two-Person Zero-Sum Games 7 X X -7 ZZ 7 -7 -5 X 5 6 -6 4 -4 1 -1 -2 X 2

(4) 1A, 2β, 3ii (two-person non-cooperative nonzero-sum) Classification of Games (limited to three dimensions of their differences) I Two-Person and N-Person games (1A, 1B) Cooperative and non-cooperative games (2ά, 2β) Zero-sum and nonzero-sum games (3i, 3ii) From combinatorial point of view, eight combinations of attributes (types of games) are possible: (1) 1A, 2ά, 3i (two-person cooperative zero-sum) (2) 1A, 2ά, 3ii (two-person cooperative nonzero-sum) (3) 1A, 2β, 3i (two-person non-cooperative zero-sum) (4) 1A, 2β, 3ii (two-person non-cooperative nonzero-sum) (5) 1B, 2ά, 3i (N-person cooperative zero-sum) (6) 1B, 2ά,3ii (N-person cooperative nonzero-sum) (7) 1B, 2β, 3i (N-person non-cooperative zero-sum) (8) 1B, 2β, 3ii (N-person non-cooperative nonzero-sum)

Classification of Games (limited to three dimensions of their differences) II (1) 1A, 2ά, 3i (two-person cooperative zero-sum) This combination is inconsistent (contradictory): in antagonistic conflict between two players, there is no ground for cooperation However, if there are at least 3 players and the situation is zero-sum, it is possible for some of them (two or more) to get payoffs at the expense of others, so there is ground for cooperation: coalition of two (or more) against the other(s) Generally, the N-person (N>2) game theory is the analytical tool of the coalition theory. For other applications of game theory, the knowledge of 2-person games is sufficient. In Coleman’s book, at some places 3-person games are used. Among them (4) 1A, 2β, 3ii (non-cooperative nonzero-sum) games are most interesting and widely applicable for the construction of models in social theory and research