Dynamic Contest and Cooperation�

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Roman Sheremeta, Ph.D.

Professor, Weatherhead School of Management

Case Western Reserve University

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Outline�

• Review
• Dynamic contests and cooperation

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Review:�Infinitely Repeated Game

• Stage game:
• G = {S₁, S₂; u₁, u₂} denotes a static stage game in which player 1 chooses an action s₁ from the action space S₁ and player 2 chooses an action s₂ from the action space S₂, and payoffs are u₁ and u₂, respectively

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• Infinitely repeated game:
• Given a stage game G, let G(∞) denote the infinitely repeated game in which G is played infinitely number of times, with the outcomes of all preceding plays observed before the next play begins
• In other words, an infinitely repeated game is a dynamic game of complete information in which a simultaneous-move game is played infinitely many times

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Review:�Infinitely Repeated Game

• Rules of the game:
• The simultaneous-move game is played at stage 1, 2, 3, ..., t-1, t, t+1, .....
• The outcomes of all previous t-1 stages are observed before the play at the t^th stage
• The stream of payoffs π₁, π₂, π₃, …. is discounted by a factor δ, where 0 < δ < 1
• Present value payoff = π₁ + δπ₂ + δ²π₃ + δ³π₄ + …

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• Some common strategies:
• Always defect
• Always cooperate
• Tit-for-tat
• Grim trigger

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Review:�Infinitely Repeated Game

• To check if a strategy is a Markov perfect equilibrium, we need to check if the combination of strategies is a Nash equilibrium of the infinitely repeated game

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Contest vs Cooperation�

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Contest�

• The normal-form representation:

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• Set of players: {Player 1, Player 2}
• Sets of strategies: S₁=[0, +∞), S₂=[0, +∞)
• Payoff functions: u₁(x₁,x₂) = Vx₁/(x₁+x₂) - cx₁

u₂(x₁,x₂) = Vx₂/(x₁+x₂) - cx₂

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Contest:�Non-Cooperative

• Suppose that the two players act non-cooperatively by maximizing their individual payoffs

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• The payoff functions are: u₁(x₁,x₂) = Vx₁/(x₁+x₂) - cx₁ u₂(x₁,x₂) = Vx₂/(x₁+x₂) - cx₂

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• Solution to this problem is: x₁* = (Vx₂/c)^0.5 - x₂ x₁^NC = x₂^NC = V/(4c) u₁^NC = u₂^NC = V/4

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Experiment #3: �Contest

• In the class experiment:
• The prize value V = 100
• The cost parameter is c = 1

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• Given these parameters, at Nash equilibrium:
• Individual output: x₁* = x₂* = 100/4 = 25
• Individual firm’s profit: u_i(x₁*, x₂*) = 100/4 = 25

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Experiment #3: �Results (2019 CWRU)

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Contest:�Cooperative

• Suppose that the two players act cooperatively by maximizing their joint payoff

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• The joint payoff is: u₁(x₁, x₂) + u₂(x₁, x₂) = = Vx₁/(x₁+x₂) - cx₁ + Vx₂/(x₁+x₂) - cx₂ = = V - cx₁ - cx₂ = V - c{x₁+x₂} = V - cx, where x = x₁+x₂

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• Solution to this problem is: x* = 0 x₁^C = x₂^C = 0 u₁^C = u₂^C = V/2

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Contest:�Grim Trigger Strategy

• Grim trigger strategy:
• Start by cooperating and continues to do so as long as the other player cooperates
• If the other player defects than use non-cooperative strategy forever

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• When is the grim trigger strategy an equilibrium?
• Suppose that player 1 plays the grim trigger strategy
• Suppose that player 2 plays the grim trigger strategy up to stage t-1
• Can player 2 be better-off if she deviates from the grim trigger strategy at stage t?

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Contest:�Grim Trigger Strategy

• Case 1: Play the trigger strategy
• If player 2 continues to play the trigger strategy x₂^C at stage t and after, then she will get a sequence of payoffs u₂^C = V/2 from stage t to stage +∞
• u₂^C + u₂^Cδ + u₂^Cδ² + … = u₂^C/(1− δ) = = V/2(1− δ)

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Contest:�Grim Trigger Strategy

• Case 2: Deviate from the trigger strategy
• If player 2 deviates from the trigger strategy x₂^C at stage t then she will choose x₂* = ε and receive a payoff u₂^* = V - cε
• In response, player 1 will play x₁^NC after stage t forever
• In response, player 2 should play x₂^NC forever
• So, player 2 will get a sequence of payoffs u₂^*, u₂^NC, u₂^NC, u₂^NC ... from stage t to stage +∞
• u₂^* + u₂^NCδ + u₂^NCδ² + … = u₂^* + u₂^NCδ/(1− δ) = = V - cε + Vδ/4(1− δ)

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Contest:�Grim Trigger Strategy

• Comparing the two cases:
• Case 1: If player 2 plays the trigger strategy, her payoff is: V/2(1− δ)
• Case 2: If player 2 deviates, her payoff is: V - cε + Vδ/4(1− δ)
• Hence, if V/2(1− δ) ≥ V + Vδ/4(1− δ), player 2 cannot be better off if she deviates from the trigger strategy
• Thus, if player 1 plays the trigger strategy the player 2’s best response is the trigger strategy for δ ≥ 2/3
• By symmetry, if player 2 plays the trigger strategy then the player 1's best response is the trigger strategy

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Contest:�Grim Trigger Strategy

• In the one-shot Contest game cooperation is not possible

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• However, if the game is infinitely repeated then there is a Markov equilibrium in which both players play the grim trigger strategy if future is sufficiently important, δ ≥ 2/3

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• Cooperation is good for you!

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Cooperation�

• The benefits of cooperation
• https://www.youtube.com/watch?v=jop2I5u2F3U

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Games For Fun�

• Next time we will play games in the class!
• Please search online for an interesting game (be creative!) and be ready to present the rules of your game to the entire class
• Important: The games have to be of complete and perfect information

Thank you!

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Roman Sheremeta, Ph.D.

Professor, Weatherhead School of Management

Case Western Reserve University

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References�

• Watson, J. (2013). Strategy: An Introduction to Game Theory (3rd Edition). Publisher: W. W. Norton & Company. (Chapter 23)

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