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Game Theory

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Roman Sheremeta, Ph.D.

Professor, Weatherhead School of Management

Case Western Reserve University

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Part 1: Game Theory

Part 2: Nash Equilibrium

Part 3: Application to Collective Action

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Part 1: �Game Theory

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What is a game?�

  • Game: A situation where a group of people is affected by the choices made by other individuals within that group.

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  • Essential elements of a game:
    1. Set of players (at least two).
    2. Set of actions for each player.
    3. Preferences over the set of actions.
    4. Strategies – a complete plan of actions.

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  • Strategy:
    • A strategy is a part of the mental skill needed to play well.
    • How best to make a choice that will maximize utility, anticipating the choice of the opponent.

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Game theory�

  • Game theory: A mathematical approach to modeling behavior by analyzing the strategic decisions made by interacting rational players.

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  • Assumptions:
    • Players are perfectly rational.
    • Maximize expected utility.
    • Care only about monetary incentives.
    • Completely selfish.

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Why game theory is important?�

  • Games are a convenient way to model strategic interactions among agents:
    • Labor: Receiving a promotion depends not only on your effort but also on efforts of others.
    • Industrial organization: Price strategy depends not only on your output but also on the output of your competitor.
    • Public goods: My benefits from contributing to a public good depend on what everyone else contributes.

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  • Game theory in other disciplines:
    • Political science: Elections, voting, conflicts, wars.
    • Computer science: Multi-agent systems, algorithms.
    • Law: Arbitration, litigation.
    • Biology: Animals fighting for dominance.

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Part 2: �Nash Equilibrium

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Nash equilibrium�

  • Nash equilibrium is a set of strategies, one for each player, such that each player’s strategy is best for her, given that all other players are playing their best strategies (Nash 1950).
    • In other words, Nash equilibrium is a stable situation that no player would like to deviate if others stick to it.

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  • John F. Nash
    • 1994 Nobel Prize in Economics
    • “Beautiful Mind”

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Example�

  • Strategy profile (B, R) has the following property:
    • Player 1 cannot do better by choosing a strategy different from B, given that Player 2 chooses R.
    • Player 2 cannot do better by choosing a strategy different from R, given that Player 1 chooses B.
    • Therefore, (B, R) is a Nash equilibrium: each player’s strategy is best for her, given that all other players are playing their best strategies.

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Player 2

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L

C

R

Player 1

T

0 , 5

5 , 0

3 , 3

M

5 , 0

0 , 5

3 , 3

B

3 , 3

3 , 3

4 , 4

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Using best response�

  • Best response:
    • What is Player 1’s best response if Player 2 chooses L?
    • What is Player 1’s best response if Player 2 chooses C?
    • What is Player 1’s best response if Player 2 chooses R?
    • What is Player 2’s best response if Player 1 chooses T?
    • What is Player 2’s best response if Player 1 chooses M?
    • What is Player 2’s best response if Player 1 chooses B?
    • Therefore, (B, R) is a Nash equilibrium: no player wants to deviate.

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Player 2

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L

C

R

Player 1

T

0 , 5

5 , 0

3 , 3

M

5 , 0

0 , 5

3 , 3

B

3 , 3

3 , 3

4 , 4

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Part 3: �Application to Collective Action

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Prisoners’ Dilemma�

  • Best response:
    • What is Prisoner 1’s best response if Prisoner 2 chooses NC?
    • What is Prisoner 1’s best response if Prisoner 2 chooses C?
    • What is Prisoner 2’s best response if Prisoner 1 chooses NC?
    • What is Prisoner 2’s best response if Prisoner 1 chooses C?
    • Therefore, (C, C) is a Nash equilibrium in Prisoners’ Dilemma.

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C

NC

C

NC

-1 , -1

-9 , 0

0 , -9

-6 , -6

Prisoner 1

Prisoner 2

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Prisoners’ Dilemma�

  • Collective action problem:
    • Both players are better off as a group if they choose (A, A). Then both would receive (4,4).
    • But each player has an incentive to choose B, resulting in (1,1).
    • Thus, the collective action problem: conflict between the individual interest and the group interest results in an inferior group outcome.

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B

A

B

A

4 , 4

0 , 5

5 , 0

1 , 1

Player 1

Player 2

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Public goods�

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Public goods�

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Common goods�

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Common goods�

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References�

  • Nash, J.F. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, 36, 48-49.
  • Ostrom, E., Gardner, R., Walker, J. (1994). Rules, games, and common-pool resources. University of Michigan Press.

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