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Game theory

Jürgen ScheffranCliSAP Research Group Climate Change and Security

Institute of Geography, Universität Hamburg

Models of Human-Environment InteractionLecture 7, June 03, 2015

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Basic types of dynamic mathematical models

x(t): System state at time t

Dx(t) = x(t+1)-x(t): System change at time t

Dx(t) = f(x,t): dynamic system

Dx(t) = f(x,u,t): dynamic system with control variable u

Dx(t) = f(x,u1,u2,t): dynamic game with control variables u1,u2 of two agents 1 and 2

Dx(t) = f(x,u1,…un,t): agent-based model and social network with control variables u1,…un of multiple agents 1,…,n

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What is game theory?

Formal method to analyse rational decisionmaking in social conflict situations in which the success of actors and decisionmakers depends on not only on the own decisions but also on the decisions of aother actors.

Special case of the general decision theory where the environmental situations depend on other actors.

Decisions in conflict situations, in which there is no control over other decision units.

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Game situation

The result depends on the decision of several players such that a single player cannot determine the outcome independent of other players.

Each player is aware of this interdependence and can assume that the others are also aware of this interdependence which is considered in all decisions.

Each players knows the strategies of all players and the results of all game situations.

Each player has a preference order regarding the results which can be subject according the transitive comparison of two results and is known to the competitors.Decisions of the player are consistent with their goals (``rational behavior'') and follow given rules

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Classification of games

Number of players: One-person game (against nature), two-player game (bipolar), $N$-player game (multipolar)

Type of player: Single person, groups, firms, nations. Degree of information: perfect information (chess), imperfect information

(Poker) Comparability of values/goals: Zero-sum game, non-zero sum game Type of players: pure strategy (deterministic), mixed strategy (probabilistic) Form of strategy set: discrete vs. continuous game, finite vs. infinite game Representation form: extensive form (tree), normal form

(matrix),characteristic function form. Type of solution concept: dominant strategy, saddle point strategy

(Minimax), Nash equilibrium, Pareto optimum,... Degree of cooperation: non-cooperative game (independence of players),

cooperative games (communication, negotiation, agrement, distribution of gains) Form of dynamics: static games, dynamic games, differential games,

evolutionary games

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Stages in the development of game theory

• B. Pascal to P. Fermat (29. Juli 1654): Origin of probability theory• Waldegrave (1712): Discovery of optimal mixed strategies for the game `Le Here'• D. Bernoulli (1732): Analysis of the St. Petersburg games; idea of a rational decision

finding• P. Laplace (1814): Consideration of the optimality principle• J. Bertrand (1888): game theoretic approach to game Baccarat• E. Zermelo (1911): game theoretic approach to chess• E. Borel (1912, 1921): systematic study of matrix games, existence of optimal mixed

strategies• J. von Neumann (1928): ``Zur Theorie der Gesellschaftsspiele''; essential ideas of

game theory • J. von Neumann, O. Morgenstern (1944): ``Theory of Games and Economic

Behaviour''• A. Tucker (1950): ``Discovery'' of the prisoners dilemma• J. Nash (1950): non-cooperative game theory; existence of equilibria• J. Nash, L. Shapley (1953): Foundation of cooperative game theory• D. Luce, H. Raiffa (1957): ``Games and Decisions''; stochastic and repeated games• T.C. Schelling (1960): ``The Strategy of Conflict''• J.C. Harsanyi (1967): Concept of incomplete information• A. Rapoport, R. Axelrod (1984): Tit-for-Tat• J.C. Harsanyi, J.F. Nash, R. Selten (1994): Nobel price for economics

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John Nash

John Forbes Nash, Jr. * 13. June 1928 in Bluefield, West Virginia; † 23. May 2015, New JerseyUS-American mathematician in game theory, differential geometry, partial differential equations.1994 Nobel Prize in Economics for gametheory with Reinhard Selten & John Harsanyi.

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Definition of games

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x0

x2

e4 e5

e6 e7x5

x3

x1

x4

e3

e2e1(5,2) (0,5)

(1,1)

(6,2) (0,0)

(10,-10) (-10,10)

1

1

2

2

2/3

1/2

2 1

1/3

21

1/2

P1

P2

P1

P2

P0

P1

Extended game form

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Normal form of the Matrix game

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Solution concepts

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Solution concepts

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Nash equilibrium

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Pareto optimum

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The Prisoners Dilemma

Two prisoners and the sheriff: Defect or not defect?

Prisoners can blame each other (defect)

Prisoners can both keep quiet (cooperate)

Preference order for each prisoner:1. Unilaterally defect to reduce punishment

2. Both cooperate

3. Both defect

4. Cooperate while the other defects

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The Prisoners Dilemma game

Cooperate

Defect

Cooperate

(2,2)

(3,3)

(4,1)

(1,4)

Defect

Preference order (1: best, 4: worst)

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The Prisoners Dilemma game (dynamic)

Cooperate

Defect

Cooperate

(2,2)

(3,3)

(4,1)

(1,4)

Defect

Preference orderNash equilibrium bold

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Prisoners Dilemma in international security

Game setting• Two countries can cooperate and disarm• One country can arm for its own security • One country cannot solely guarantee security

Preference order for each country1. Arm while the other disarms

2. Mutual cooperation and disarmament

3. Both arm

4. Disarm while the other country arms

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The Prisoners Dilemma gamein international security (dynamic)

Disarm

Arm

Disarm

(2,2)

(3,3)

(4,1)

(1,4)

Arm

Preference order

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The Stag Hunt Game

Hunting stag or hare?

Hunters can hunt hare individually

Stag is more valuable, but can be hunted only jointly

Preference order for each hunter1. Both hunt stag

2. Hunt hare while other hunts stag

3. Both hunt hare

4. Hunt stag while other hunts hare

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The Stag Hunt Game

Stag

Hare

Stag

(1,1)

(3,3)

(4,2)

(2,4)

Hare

Preference order

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Stag Hunt game (dynamic)

Stag

Hare

Stag

(1,1)

(3,3)

(4,2)

(2,4)

Hare

Preference orderNash equilibria bold

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Stag Hunt game and international security

Game setting• Two countries can cooperate and disarm• One country can arm for its own security • One country cannot solely guarantee security

Preference order for each country1. Mutual cooperation and disarmament

2. Arm while the other disarms

3. Both arm

4. Disarm while the other country arms

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Stag Hunt game in international security

Disarm

Arm

Disarm

(1,1)

(3,3)

(4,2)

(2,4)

Arm

Preference orders

USSR and US during the Cold War

USA

USSR

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Stag Hunt game in international security (dynamic)

Disarm

Arm

Disarm

(1,1)

(3,3)

(4,2)

(2,4)

Arm

Preference ordersNash equilibria bold

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The Chicken game

No attack

Attack

No attack

(2,2)

(4,4)

(3,1)

(1,3)

Attack

Preference orderNash equilibria bold

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Chicken games in culture and politics

James Dean, Rebel Without a Cause (1955)

Cuban Missile Crisis1962

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The Chicken game (dynamics)

No attack

Attack

No attack

(2,2)

(4,4)

(3,1)

(1,3)

Attack

V(A,N) > V(N,N) > V(N,A) > V(A,A)V(N,A) > V(N,N) > V(A,N) > V(A,A)

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How to address the security dilemma?

Increase the incentives to cooperate (reward)

Decrease incentives for defecting (punishment)

Increase trust that both will cooperate (verification)

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The inspection game (dynamics)

Accept

Alarm

Comply

(1,2)

(3,4)

(4,1)

(2,3)

CheatInspector

Inspected

Preference orderNo Nash equilibria: circular behavior

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Option 3 increases complexity

a1

a3

a2

b2b1 b3

(3,3) (0,4)

(1,5)

(2,2)

(4,1)

(3,0)

(1,4) (5,1)

(4,4)

Numbers represent values (not preferences)

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Multiple cooperative equilibria

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Mixed strategies

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Repeated games

General conclusion:• It is better to be nice than evil.• It is important to be reactive.• It is important to forgive fast..• Deceit does not pay.

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Results of Axelrod’s computer competition

Source: Axelrod 1984