Rational Choice Rational Choice SociologySociology

Lecture 5Lecture 5

Game theory I: the concept Game theory I: the concept and classification of gamesand classification of games

The Concept of Game IThe Concept of Game I Game theory is the branch of RCT that Game theory is the branch of RCT that

analyzes the problem of rational choice in analyzes the problem of rational choice in strategic situation; or: theory of strategic situation; or: theory of strategically rational actionstrategically rational action

Strategic situation is the situation of choice Strategic situation is the situation of choice involving at least two rational actors making involving at least two rational actors making interdependent choices and knowing about interdependent choices and knowing about this interdependencethis interdependence

Interdependence means that outcomes of Interdependence means that outcomes of alternative choices of each actor depend on alternative choices of each actor depend on what other actors choosewhat other actors choose

Game is analytical model of a strategic Game is analytical model of a strategic situationsituation

The Concept of Game IIThe Concept of Game II

Actor2Actor2

(Columns)(Columns)

Actor 1Actor 1

(Rows)(Rows)

C1C1 C2C2Action 1Action 1 rr1111 rr1212

Action 2Action 2 rr2121 rr2222

ActioAction1n1

ActioAction2n2

ActioAction1n1 rr1111 rr1212

ActioAction2n2 rr2121 rr2222

The Concept of Game IIIThe Concept of Game III The parametric situations can be considered as The parametric situations can be considered as involvinginvolving the the

game against nature (including people behaving like naturegame against nature (including people behaving like nature))

Strategic actions can be considered as the subclass of the Strategic actions can be considered as the subclass of the social actions in the sense of Max Weber. Social action: social actions in the sense of Max Weber. Social action: action oriented to the actions of other actors. Strategic action oriented to the actions of other actors. Strategic action is instrumentally rational social action in the strategic action is instrumentally rational social action in the strategic situationsituation

The social actions of at least two actors constitute a game The social actions of at least two actors constitute a game (or can be de(or can be desscribed as game) if following conditions are cribed as game) if following conditions are satisfied:satisfied:

(1)(1) Each actor is instrumentally rational (maximizes utility or Each actor is instrumentally rational (maximizes utility or expected utility): this means that her preferences are expected utility): this means that her preferences are consistent (satisfy axioms discussed above)consistent (satisfy axioms discussed above)

(2)(2) Each actor knows (1): assumption of symmetryEach actor knows (1): assumption of symmetry(3)(3) Each actor knows that other actors know that she knows (2): Each actor knows that other actors know that she knows (2):

= assumption of the common knowledge of rationality= assumption of the common knowledge of rationality(4)(4) From the same information each actor makes the same From the same information each actor makes the same

conclusions (no misperceptions or misunderstandings): = conclusions (no misperceptions or misunderstandings): = assumption of common alignment of beliefsassumption of common alignment of beliefs

The Concept of Game IVThe Concept of Game IVThe description of a game includes the specification of:The description of a game includes the specification of: Actor set N {1,2,.., n}, including at least two playersActor set N {1,2,.., n}, including at least two players Strategy set S {1,2, …, m}, including at least two Strategy set S {1,2, …, m}, including at least two

strategiesstrategies Outcome set O. The size of this set depends on how Outcome set O. The size of this set depends on how

many players there are and how many strategy many players there are and how many strategy choices each of them has.choices each of them has.

E.g. if there are two players and each of them has 2 E.g. if there are two players and each of them has 2 strategies, there strategies, there are are 2×2= 4 outcomes; if there are 2×2= 4 outcomes; if there are two players, one of them has 2, another 3 strategies, two players, one of them has 2, another 3 strategies, there are 2×3=6 outcomes; if there are 3 players, one there are 2×3=6 outcomes; if there are 3 players, one of them has 2, another 4, and the third one 6 of them has 2, another 4, and the third one 6 strategies, there are 2×4×6=48 outcomes etc.strategies, there are 2×4×6=48 outcomes etc.

Names of the outcomes can be built using Names of the outcomes can be built using as elements as elements the names the names of strategies that cause themof strategies that cause them

Utility functions U that for each player and for each Utility functions U that for each player and for each outcome ascribe the utility index u (ordinal or cardinal)outcome ascribe the utility index u (ordinal or cardinal)

The Concept of Game VThe Concept of Game V

Actor 2Actor 2

Actor 1Actor 1

Actor 1: sActor 1: s22ss11 >> s s11ss11 >> s s22ss22 >>ss11 s s22

4 3 2 14 3 2 1

Actor 2: sActor 2: s11ss22 >> s s11ss11 >> s s22ss22 >> s s22ss11

4 3 2 14 3 2 1

ss11 ss22

ss11 3 s3 s11ss1 1 33 11 ss11ss2 2

44

ss22 4 s4 s22ss11 11

2 s2 s22ss2 2

22

The Concept of Game VIThe Concept of Game VI

Actor 2Actor 2

Actor 1Actor 1

Actor 1: sActor 1: s22ss11 >> s s11ss11 >> s s22ss22 >>ss11 s s22

22(4) 20(3) 14(2) 3(1)22(4) 20(3) 14(2) 3(1)

Actor 2: sActor 2: s11ss22 >> s s11ss11 >> s s22ss22 >> s s22ss11

-2(4) -4(3) -30(2) -55(1)-2(4) -4(3) -30(2) -55(1)

ss11 ss22

ss11 20 s20 s11ss1 1 -4-4 33 ss11ss2 2 -2-2

ss2222 s22 s22ss11 -55 -55 14 s14 s22ss2 2 -30-30

The Concept of Game: an The Concept of Game: an ExampleExample

(Prisoner’s Dilemma)(Prisoner’s Dilemma) BillBill

John John

The police have caught two suspected drug dealers, John and Bill. They are now sitting 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 in two separate cells in the police station and cannot communicate. The prosecutor tells them that they have one hour to them that they have one hour to decide confess decide confess their crimes or deny the charges. The their crimes or deny the charges. The prosecutor has enough evidence to prove the charge to sentence them for 2 years for a prosecutor has enough evidence to prove the charge to sentence them for 2 years for a well-documented offences. However, to sentence themwell-documented offences. However, to sentence them for drug trafficking the for drug trafficking the prosecutor should have the confession of at least one of them. The legal situation is as 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 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 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.with just 1 year in prison, whereas the other will get 10 years.

Do not Do not confessconfess

(cooperate)(cooperate)

Confess Confess

(defect)(defect)

Do not confessDo not confess

(cooperate)(cooperate) -2 s-2 s11ss1 1 -2-2 -10-10 ss11ss2 2 -1-1

Confess Confess

(defect)(defect)-1 s-1 s22ss11 -10 -10 -8 s-8 s22ss2 2 -8-8

Taxonomy (classification) of Taxonomy (classification) of gamesgames

Two-person and N-person (N>2) Two-person and N-person (N>2) ((How mHow manyany players) players)

Cooperative and non-cooperative gamesCooperative and non-cooperative games(in cooperative games players can agree on binding contracts that force them to (in cooperative games players can agree on binding contracts that force them to

respect whatever they have agreed)respect whatever they have agreed)

Non-iterated (one-shot) and iterated gamesNon-iterated (one-shot) and iterated games Simultaneous-move versus sequential-move gamesSimultaneous-move versus sequential-move games Games with perfect information versus games with Games with perfect information versus games with

imperfect informationimperfect information(sequential game is a game with perfect information if players have full information about (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 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 about their previous moves, the sequential game is with imperfect information)information)

Constant (including zero) sum games versus non-Constant (including zero) sum games versus non-constant sumconstant sum

Two-Person Constant-Sum GamesTwo-Person Constant-Sum Games

A game is constant-sum if for all outcomes A game is constant-sum if for all outcomes the sum of payoffs for all players is the sum of payoffs for all players is constant.constant.

A special case of constant-sum game is A special case of constant-sum game is zero-sum game where this sum is zero. zero-sum game where this sum is zero.

All constant-sum games sometimes are All constant-sum games sometimes are called zero-sum, because each constantcalled zero-sum, because each constant--sum game can be transformed into a zerosum game can be transformed into a zero--sum game by subtracting from each sum game by subtracting from each payoff half of the sum of the payoffspayoff half of the sum of the payoffs

Two-person zero-sum gameTwo-person zero-sum game

ss11 ss22 ss33

ss112 32 3 4 14 1 5 05 0

ss222 32 3 1 41 4 2,5 2,52,5 2,5

ss330 50 5 4 14 1 0 50 5

ss11 ss22 ss33

ss11-0,5X-0,5XXX 0,50,5ZZZZ

1,5 -1,51,5 -1,5ZZ 2,5 -2,52,5 -2,5ZZ

ss22--0,5 0,50,5 0,5ZZ -1, 5-1, 5XX 1,5 1,5 0 00 0

ss33--2,52,5XX 2,5 2,5 1,5 -1,5 -

1,51,5ZZ--2,52,5XX 2,5 2,5

Two-Person Zero-Sum Two-Person Zero-Sum GamesGames Zero-sum games are models of so-called antagonistic Zero-sum games are models of so-called antagonistic

conflicts (conflicts (e.g. e.g. war). For each outcome, one player can war). For each outcome, one player can gain only as much the other player loose.gain only as much the other player loose.

Zero-sum games is the oldest province of the game Zero-sum games is the oldest province of the game theory. They were investigated intheory. They were investigated in an an exhaustive way exhaustive way in the book J. von Neumann and O. Morgenstern in the book J. von Neumann and O. Morgenstern Theory of Games and Economic Behavior (1944)Theory of Games and Economic Behavior (1944)

In this book, the authors proved that in the zero-sum In this book, the authors proved that in the zero-sum games the choice according to minimax(maximin) rule games the choice according to minimax(maximin) rule is rational (utility-mis rational (utility-maximizingaximizing) for all player) for all playerss

A pair of strategies are in equilibrium if the A pair of strategies are in equilibrium if the outcome determined by the strategies equals outcome determined by the strategies equals the minimal value of the the minimal value of the row and the maximal row and the maximal value of the value of the column (minimax conditioncolumn (minimax condition rule rule))

Two-Person Zero-Sum Two-Person Zero-Sum GamesGames

ss11 ss22 ss33

ss119 - 99 - 9 Z Z 8 -88 -8

ZZ7 X 7 X XX -7 -7 ZZ ZZ

ss227 -77 -7 -5 -5 XX 5 5 6 -66 -6

ss334 -44 -4 1 -11 -1 -2 -2 XX 2 2

Classification of Games (limited to three Classification of Games (limited to three dimensions of their differences)dimensions of their differences) I I

Two-Person and N-Person games (1A, 1B)Two-Person and N-Person games (1A, 1B) Cooperative and non-cooperative games (2Cooperative and non-cooperative games (2άά, 2, 2ββ)) Zero-sum and nonzero-sum games (3i, 3ii)Zero-sum and nonzero-sum games (3i, 3ii)From combinatorial point of view, eight combinations From combinatorial point of view, eight combinations

of attributes of attributes (types of games) are possible:(types of games) are possible:(1) (1) 1A1A, , 22άά, , 3i3i (two-person cooperative zero-sum) (two-person cooperative zero-sum)((22) ) 1A, 21A, 2άά, 3ii, 3ii ( (two-person cooperative nonzero-sum)two-person cooperative nonzero-sum)(3)(3) 1A, 1A, 22ββ, 3i, 3i ( (two-person non-cooperative zero-sum)two-person non-cooperative zero-sum)

(4)(4) 1A, 1A, 22ββ, 3ii, 3ii ( (two-person non-two-person non-cooperative nonzero-sum)cooperative nonzero-sum)

(5)(5) 1B, 21B, 2άά, 3i, 3i ( (N-N-ppersonerson cooperative zero-sum) cooperative zero-sum)(6)(6) 1B, 21B, 2άά,3ii,3ii (N-person (N-person cooperative nonzero-sum)cooperative nonzero-sum)(7) (7) 1B, 1B, 22ββ, 3i, 3i (N-person (N-person non-cooperative zero-sum)non-cooperative zero-sum)(8)(8) 1B, 1B, 22ββ, 3ii, 3ii (N-person (N-person non-cooperative nonzero-non-cooperative nonzero-

sum)sum)

Classification of Games (limited to Classification of Games (limited to three dimensions of their differences)three dimensions of their differences) II II

((11) ) 1A, 21A, 2άά, 3i, 3i ( (two-person cooperative zero-sum)two-person cooperative zero-sum)

This combination is inconsistent (contradictory): in This combination is inconsistent (contradictory): in antagonistic conflict between two players, there is no antagonistic conflict between two players, there is no ground for cooperationground for cooperation

However, if there are at least 3 players and the situation is 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 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 get payoffs at the expense of others, so there is ground for cooperation: coalition of two (or more) against the other(s)cooperation: coalition of two (or more) against the other(s)

Generally, the N-person (N>2) game theory is the analytical Generally, the N-person (N>2) game theory is the analytical tool of the coalition theory. For other applications of game tool of the coalition theory. For other applications of game theory, the knowledge of 2-person games is sufficient. In theory, the knowledge of 2-person games is sufficient. In Coleman’s book, at some places 3-person games are used. Coleman’s book, at some places 3-person games are used.

Among them Among them (4)(4) 1A, 1A, 22ββ, 3ii, 3ii ( (non-cooperative nonzero-sum) non-cooperative nonzero-sum) gamesgames

are most interesting and widely applicable for the are most interesting and widely applicable for the construction of models in social theory and researchconstruction of models in social theory and research