Incentives and Behavior Prof. Dr. Heiner Schumacher KU Leuven 1. - - PowerPoint PPT Presentation

Incentives and Behavior

• Prof. Dr. Heiner Schumacher

KU Leuven

• 1. Game Theory I
• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Introduction

Game Theory is a mathematical language that is used in social sciences to analyze strategic interaction. We have a situation of strategic interaction when the behavior of two

• r more parties in‡uences each party’s well-being.

There is an in…nite number of examples for strategic interactions: competition between …rms, the interaction between …rms and customers, auctions, bargaining, etc. This chapter introduces the basic notation and analyzes static games under complete information.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Introduction

Overview Fundamental Concepts Nash Equilibrium Mixed Strategies

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Fundamental Concepts

De…nition of a game: A number of players i 2 N = f1, ..., ng. A number of possible strategies for each player: si 2 Si. A strategy pro…le is a vector s = (s1, s2, ..., sn) 2 S = S1 S2 ... Sn. The vector si denotes the vector of strategies of all players except player i, i.e., si = (s1, ..., si1, si+1, ..., sn). A utility function that determines the players’ payo¤s as a function

• f the chosen strategies, ui : S ! R (player i’s Bernoulli utility

function). A nplayer game is a list G = fN, S1, ..., Sn; u1, ..., ung which speci…es the set of players, the strategies available to each player and the utilities of each player.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Fundamental Concepts

Players need not to act simultaneously. Each player only cares for her own payo¤. Each player knows the whole structure of the game. In a game with complete information each player’s payo¤ function is common knowledge among all players.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Fundamental Concepts

When there are only two players, we can display the game in a payo¤

• matrix. For example, consider the following game with two players

S1, S2 and two strategies for each player. S1/S2 Left Right Up 1,3 0,1 Down 2,1 1,0 Interpretation: If S1 chooses “Down” and S2 chooses “Right”, then S1 receives the payo¤ 1 and S2 the payo¤ 0.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Fundamental Concepts

Dominant Strategies Consider the game on the last slide. What is the optimal strategy for player S1? If S2 chooses “Left”, then “Down” is better than “Up”. If S2 chooses “Right”, then “Down” is better than “Up”. It is therefore a dominant strategy for S1 to play “Down”. What would be the optimal strategy for S2? If S1 chooses “Up”, then “Left” is better than “Right”. If S1 chooses “Down”, then “Left” is better than “Right”. It is therefore a dominant strategy for S2 to play “Left”. A dominant strategy is a strategy which is always optimal, that is, regardless of the opponents’ strategies. In the example above, we get an equilibrium in dominant strategies.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Fundamental Concepts

Iterated elimination of strictly dominated strategies (IESDS) In most games, there are no dominant strategies (and therefore no equilibrium in dominant strategies). How can we analyze such games? Consider the following one. S1/S2 Left Middle Right Up 1,0 1,2 0,1 Down 0,3 0,1 2,0 Idea: We drop strictly dominated strategies from the payo¤ matrix. A strategy s0

i is strictly dominated by strategy si if strategy si yields a

strictly larger payo¤ than strategy s0

i for all possible choices of the

• pponents: ui(s0

i , si) < ui(si, si) 8si 2 Si.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Fundamental Concepts

We apply the method of “iterated elimination of strictly dominated strategies” to the game on the previous slide. “Right” is strictly dominated by “Middle”. Thus, a rational S2 will never choose “Right”. If S1 knows that S2 is rational, she can eliminate “Right” from the payo¤ matrix. In this case, “Down” is strictly dominated by “Up”. If S2 knows that S1 is rational and that S1 knows that S2 is rational, she can eliminate “Down” from the payo¤ matrix. In this case, “Left” is strictly dominated by “Middle”. Conclusion: The outcome of the game is “Up, Middle”.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Fundamental Concepts

IESDS requires that the rationality of the players is common knowledge: All players are rational; All players know that all the players are rational; All the players know that all the players know that all the players are rational; asf. The order of elimination does not matter (you may try to show this as an exercise). Do not eliminate weakly dominated strategies (they are not ruled out by common knowledge of rationality).

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Nash Equilibrium

The previous solution concepts are not helpful for most games of

• interest. Consider the following example.

S1/S2 Left Middle Right Up 3,3 2,4 3,2 Middle 4,2 6,6 0,8 Down 2,3 8,0 4,4 In this game, there are neither dominant, nor strictly dominated strategies. We therefore search for the best responses. The strategy si is a best response to si if ui(si, si) ui(s0

i , si) 8s0 i 2 Si.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Nash Equilibrium

Mark for each strategy of S2 the best response of S1 by underlining the corresponding payo¤ of player S1. Then mark for each strategy of S1 the best response of S2 by underlining the corresponding payo¤ of player S2. If there are more than one maximal payo¤s in a column (row), mark all of them. We have a Nash equilibrium if there is a cell where both payo¤s are

• underlined. The corresponding strategies are “mutual best responses”.
• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Nash Equilibrium

The strategy pro…le (s

1 , ..., s n ) is a Nash equilibrium if, for each

player i and every strategy si of player i, s

i is a best response to s i,

i.e. for every player i and si ui(s

i , s i) ui(si, s i) 8si 2 Si.

Note that only a weak inequality is required. In a Nash equilibrium, no player has an incentive to unilaterally deviate (i.e., deviate from the predicted outcome when the other players play according to the prediction). Note: When he chooses his strategy, he does not know his opponent’s

• strategy. He only plays a best response against the opponent’s

expected behavior. In an equilibrium, expectations must be correct.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Nash Equilibrium

We can interpret the Nash equilibrium as follows: Suppose the players meet before the game and discuss what strategies they should play. They can only agree on strategy pro…les that are self-enforcing: each player’s strategy must be a best-response to the strategies of the

• ther players.

If players do not meet before playing the game, then in equilibrium they must have beliefs about the opponents’ strategy that are mutually consistent.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Nash Equilibrium

Why should we expect that a Nash equilibrium is played? Rational analysis of the game (which outcome makes sense?). Recommendation of a third party (examples: tra¢c signals, social norms). Communication before the game is played. Learning through trial and error.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Nash Equilibrium

Multiple equilibria In many games, there are several Nash equilibria. Consider, for example, the following game (which is called “the battle of the sexes”): “She”/“He” ballet boxing ballet 2,1 0,0 boxing 0,0 1,2 Show that both “ballet, ballet” and “boxing, boxing” are Nash equilibria!

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Nash Equilibrium

When there are several Nash equilibria, it is not clear which one will be played (or whether a Nash equilibrium will be played at all). There are some concepts that try to solve the problem. Focal Points (Schelling) Pareto optimality Coordination by communication Equilibrium re…nements

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Nash Equilibrium

Focal points

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Nash Equilibrium

Focal points are properties of the environment that can be used to anticipate the opponents’ behavior. They demonstrate that people can coordinate without communication. Schelling’s example goes as follows: Tomorrow you have to meet a stranger in New York. Where and when do you meet him?

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Nash Equilibrium

“Noon at the information booth at Grand Central Station”

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Nash Equilibrium

Pareto optimality We may argue that a particular Nash equilibrium is played if yields much higher payo¤s for the players than any other equilibrium. S1/S2 Left Right Up 100,100 0,0 Down 0,0 1,1 Here, “Up, Left” is the payo¤ dominant Nash equilibrium. However, if players are uncertain about their opponents’ strategy, it may make sense for them to choose another strategy. S1/S2 Left Right Up 9,9 0,8 Down 8,0 7,7 A Nash equilibrium is risk dominant if it is “less risky”.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Mixed Strategies

Sometimes there does not exist a Nash equilibrium in pure strategies. Consider, for example, the following game: S1/S2 Left Right Left

• 1,1

1,-1 Right 1,-1

• 1,1

There does not exist a Nash equilibrium in pure strategies. However, there exists a Nash equilibrium in mixed strategies: Each player chooses “Left” with probability 1/2 and “Right” with probability 1/2. A mixed strategy for player i is a vector that assigns to each pure strategy a probability with which this strategy will be chosen.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Mixed Strategies

How do we …nd an equilibrium in mixed strategies? Suppose that S1 expects S2 to choose “Left” with probability q. The expected payo¤ of S1 if she also chooses “Left” is q (1) + (1 q) 1 = 1 2q, and if she chooses “Right”, it is q 1 + (1 q) (1) = 2q 1. She is indi¤erent between “Left” and “Right” (i.e., both strategies yield the same expected payo¤) if and only if q = 1/2. The same holds for S2. At a mixed strategy, the player is always indi¤erent between those pure strategies that he chooses with positive probability.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Mixed Strategies

A pure strategy is a degenerate mixed strategy. For a mixed strategy to be best response to some (mixed) strategy pro…le it must put positive probability on a given pure strategy only if the pure strategy is itself a best response to the (mixed) strategy pro…le (why?). In generic games, there is an uneven number of Nash equilibria (if this number is …nite). What could be an interpretation of mixed strategies?

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Mixed Strategies

As an example, we analyze the following game. A crime is observed by a group of N people. Who reports the crime? Each person would like the police to be informed, but prefers that someone else makes the phone call. Strategies for each person: “Call” and “Don’t call”. Payo¤s. If nobody calls the police ui = 0; if i does not call the police, but somebody else: ui = v > 0; if i calls the police ui = v c > 0.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Mixed Strategies

Is there a pure strategy Nash equilibrium? Is there a symmetric pure strategy Nash equilibrium? Find the (unique) symmetric mixed strategy Nash equilibrium in which each player calls the police with probability pN 2 (0, 1). How does the probability that nobody calls the police depend on N?

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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Mixed Strategies

The game just analyzed is inspired by an event that triggered substantial research in social psychology. In March 1964, 38 individuals witnessed the murder of Catherine Genovese over a period of half an hour in New York City from their save apartments. Nobody intervened or bothered to call the police. A journalist concluded that “[i]ndi¤erence to one’s neighbor and his troubles is a conditioned re‡ex of life in New York as it is in other big cities.” However, as we just learned, the incidence can well be explained by miscoordination in the symmetric Nash equilibrium.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 1. Game Theory I

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