Game theory 9
maqi0912
week9GameTheory2.pptxMore Games
Chapter 6. Sections 6.6 to 6.9
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Thus far, we’ve covered:
Basic ingredients of games: Players, payoffs, and strategies
Dominant strategy: A strategy that is a best response to every strategy of the opponent (the other player).
E.g., Prisoners’ dilemma, exam-vs-presentation game
Nash equilibrium: When neither player has a strictly dominant strategy, players choose strategies that are best responses to each other, then no player has an incentive to deviate to an alternative strategy — so the system is in a kind of equilibrium state
Multiple equilibria: When there are more than one strategy resulting in equilibrium
Coordination games (e.g., PowerPoint vs. Keynote use game): When the multiple equilibria present equal payoffs
Unbalanced coordination game (e.g., the battle of sexes and stag hunt game): When the multiple equilibria present payoffs unequal to each player
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This class, we will cover:
Multiple equilibria (continued): Hawk-Dove game (aka the game of Chicken)
Mixed strategies:
(Thus far, all we’ve learned is “pure strategy”)
Zero sum game (when there is no equilibrium at all) how do we predict the players’ strategy choices?
Mixed strategies (given that we use probability of choosing a strategy, not binary choice of strategies)
Matching pennies, Run-pass game, Penalty-kick game
Optimality—Pareto and Social
When players reach an outcome that is good for the society
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Multiple equilibria (Continued)
The Hawk—Dove Game
- Anti-coordination game
Two Nash equilibria: (D, H) and (H, D)
Can’t predict which equilibria will be played -> No unique prediction
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The real-world application of the Hawk-Dove game
“We are both heading for the cliff, who jumps first, is the Chicken.”
Rebel without a cause
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Mixed Strategies
When there is no Nash equilibria at all, but we still want to predict what our opponent’s choice of strategy would be. What do we do?
Introduce the concept of “probability.” Then, we can always find equilibria in a game.
As opposed to “pure strategy” (binary choice), “mixed strategy” (probability)
The Matching Pennies game (one of attack-defense games, or zero-sum games)
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Mixed Strategies (Equation)
Equation=formalization=generalization
Probability of playing a strategy
p Probability of Player 1 committing to play H;
: Probability of Player committing to play T
Likewise,
q : Probability of Player 2 committing to play H;
: Probability of Player committing to play T
Expected value of the payoff
What is the definition of a Nash equilibrium again?
a pair of strategies (now probabilities) so that each is a best response to the other
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Real-world applications of mixed strategies
Tennis players trying to decide whether to serve the ball up the center or out to the side of the court;
a card-player may be randomly deciding whether to bluff or not;
two children may be randomizing among rock, paper, and scissors in the perennial elementary-school contest of the same name
…. More seriously, evolutionary biology to explain evolutionary processes (natural selection, common descent, speciation) that produced the diversity of life on Earth
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Key Concept of Mixed Strategies
Probability as opposed to binary strategy choice: Mixed Strategies
Randomize the selection of strategies
To the point where the other player feels indifferent between their two strategies.
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Run-Pass Game (Page 161)
No Nash equilibrium
Both must make their behavior unpredictable by randomizing it.
We must consider two scenarios where (1) offense make the move to make the defense indifferent between their two options, and (2) defense make the move to make the offense indifferent between the offense’s two options
The expected payoff to the offense from passing is
The expected payoff to the offense from running is
To make the offense indifferent between its two strategies, we need to set
The expected payoff to the defense from defending against the pass is:
The expected payoff to the defense from defending against the run
To make the defense indifferent between its two strategies, we need to set
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In the real-world…
The real-world data show that offense run more often than pass.
Penalty kick game (p. 163)
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Optimality
As opposed to optimizing individually, can players reach an outcome that is in any sense good (perhaps good for the society)?
Pareto Optimality
A choice of strategies — one by each player — is Pareto-optimal if there is no other choice of strategies in which every player receives a payoff at least as high (as in the payoff to be received from the other strategy, not compared to the other player’s payoff), and at least one player receives a strictly higher payoff.
All in all, this would be optimal given that at least one player is better off, and both together would not be worse off.
Wouldn’t a player switch to a different strategy?
A binding agreement can prevent such deviations. (e.g., laws, policies, and unspoken norms)
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Social Optimality
A choice of strategies — one by each player — is a social welfare maximizer (or
socially optimal) if it maximizes the sum of the players’ payoffs.
Outcome that are socially optimal must also be Pareto-optimal
But not the other way around.
Pareto Optimality
Social optimality
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Apply the two optimality concepts to the Presentation-vs-Exam game
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Going forward
I’ll attend this Thursday recitation to play Matching Pennies game against Chris
All of you are encouraged to create an account at:
https://economics-games.com/games
This site allows you to play many games (which you learned in this class) with a partner. But to do so, both you and your partner must have created login IDs. We will match you with another player during Thursday.
It will FUN this Thursday!
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