game theory - university of notre dametjohns20/marketdesign_fa16/gametheory.slide… · game theory...

Game Theory

Market Design

University of Notre Dame

Market Design (ND) Game Theory 1 / 38

Game Theory

A game is composed of

Players : Those agents who take actions

Actions or Strategies : The choices which players can select

Payoffs : The numerical value that players associate with differentoutcomes of the game, which are allowed to depend on each player’saction

Timing : A description of which players take actions when

Information : A description of what players know, and when theyknow it

Game Theory

We begin with simultaneous-move games of complete information, inwhich all players make their decisions at the same time and knoweverything about the game and about each other.

We then add timing, giving us dynamic games of completeinformation, where players make decisions in sequence, and cannotrevisit their earlier choices (think tic-tac-toe).

Finally, we add incomplete or imperfect information to get Bayesiangames (think poker).

Game Theory

A set of players i = 1, 2, ...,N

A set of feasible actions or strategies Ai for each player i

A payoff function ui (a1, a2, ..., aN) for each player i giving his payoffgiven the choices of all the players

So we can just write a game as {Ai , ui (a1, ..., aN)}Ni=1.

Game Theory

A set of players i = 1, 2, ...,N

A set of feasible actions or strategies Ai for each player i

A payoff function ui (a1, a2, ..., aN) for each player i giving his payoffgiven the choices of all the players

So we can just write a game as {Ai , ui (a1, ..., aN)}Ni=1.

Game Theory

Two more pieces of notation:

A strategy profile is a list of actions for each player:

a = (a1, a2, ..., aN),

so we write the players’ payoffs as ui (a) if a occurs.

The notationa−i = (a1, a2, ..., ai−1, ai+1, ..., aN)

is a strategy profile with the i-th player removed. It lets us focus onplayer i ’s incentives by holding a−i fixed and thinking about what ishould do:

ui (a) = ui (ai , a−i )

Game Theory

Two more pieces of notation:

A strategy profile is a list of actions for each player:

a = (a1, a2, ..., aN),

so we write the players’ payoffs as ui (a) if a occurs.

The notationa−i = (a1, a2, ..., ai−1, ai+1, ..., aN)

is a strategy profile with the i-th player removed. It lets us focus onplayer i ’s incentives by holding a−i fixed and thinking about what ishould do:

ui (a) = ui (ai , a−i )

Prisoners’ Dilemma

There are two burglars, who have been captured in the process ofcommitting a crime. They have been very careful, and actually do noteven know each other’s real name. The district attorney tells them:“If you both remain silent, I have enough evidence to send each ofyou to jail for two years. However, if one of you confesses and theother tells the truth, I will give the confessor a lighter sentence,sending him to jail for only one year, while I prosecute the otheraggressively and send him to jail for five years. If both of you confess,there won’t be a trial, and you both get three years.”

Column

Silent Confess

Row Silent -2,-2 -5,-1

Confess -1,-5 -3,-3

What are the Pareto optimal outcomes of the game? What do you thinkthe burglars do, and why? What other economic situations have similarincentives?

Battle of the Sexes

Two people have decided to go on a date. The two options are aFootball game, and the Ballet. The male prefers football, while thefemale prefers ballet. They discuss which option they will pick, butboth happen to forget which they decided on. Worse, they bothforgot their smart phones at work, and the two events are about tobegin. Both prefer to be together rather than apart.

Battle of the Sexes

Female

Football Ballet

Male Football 2,1 0,0

Ballet 0,0 1,2

What are the Pareto optimal outcomes for the couple? What do you thinkthey do, and why? What if we made the payoffs to Ballet (10, 20)? Whatother economic situations have similar incentives?

Matching Pennies

You are waiting for a plane with a friend. Both of you have plenty ofpocket change, so you propose the following game: You both secretlypick Heads or Tails. If both coins are heads, you get both coins. Ifboth coins are tails, your friend gets both coins.

Matching Pennies

Friend

Heads Tails

You Heads 1,-1 -1,1

Tails -1,1 1,-1

What are the Pareto optimal outcomes? What do you think they do, andwhy? What happens if we made the payoff to (Tails, Tails) equal to(10,−1)?

The Strategic Form

The matrix of players/actions/payoffs that we’ve been using to describegames is very helpful, since it summarizes all of the relevant informationfrom a game theory perspective. We call it the strategic form.

Column Player

Row Player U urow (U, L), ucolumn(U, L) urow (U,R), ucolumn(U,R)

D urow (D, L), ucolumn(D, L) urow (D,R), ucolumn(D,R)

So you can think of game theory as a generalization of price-taking orperfectly competitive models where consumers have preferences overbundles of goods or firms have preferences over quantities produced, to asetting where agents have preferences over how the other agents act.

The Strategic Form

The matrix of players/actions/payoffs that we’ve been using to describegames is very helpful, since it summarizes all of the relevant informationfrom a game theory perspective. We call it the strategic form.

Column Player

Row Player U urow (U, L), ucolumn(U, L) urow (U,R), ucolumn(U,R)

D urow (D, L), ucolumn(D, L) urow (D,R), ucolumn(D,R)

So you can think of game theory as a generalization of price-taking orperfectly competitive models where consumers have preferences overbundles of goods or firms have preferences over quantities produced, to asetting where agents have preferences over how the other agents act.

Best-Responses

Definition

A particular strategy a∗i is a best-response for player i to a−i if, for anyother strategy a′i that player i could choose,

ui (a∗i , a−i ) ≥ ui (a

′i , a−i )

Best-Responses

The first thing you should do when you see any game in a strategic form isto underline the players’ best responses. Consider the game:

l ru 3, ∗ −2, ∗m 2, ∗ −5, ∗d 2, ∗ −2, ∗

So u is a best-response to l , and u and d are both best-responses to r .

Best-Responses

The first thing you should do when you see any game in a strategic form isto underline the players’ best responses. Consider the game:

l ru 3, ∗ −2, ∗m 2, ∗ −5, ∗d 2, ∗ −2, ∗

So u is a best-response to l , and u and d are both best-responses to r .

Strategy Dominance

Definition

A strategy ai∗ dominates a strategy a′i for player i if, for any a−i thatplayer i ’s opponents might use,

ui (ai∗, a−i ) ≥ ui (a′i , a−i ).

Strategy Dominance

Going back to our example,

l ru 3, ∗ −2, ∗m 2, ∗ −5, ∗d 2, ∗ −2, ∗

So u dominates m and d , and d dominates m. So we’d be justified inpredicting that the row player use the strategy u.

Dominant Strategies

Definition

A strategy a∗i is a dominant strategy for player i if, for any profile ofopponent strategies a−i and any other strategy a′i that player i couldchoose,

ui (a∗i , a−i ) ≥ ui (a

′i , a−i ).

Column

Silent Confess

Row Silent -2,-2 -5,-1

Confess -1,-5 -3,-3

Example

BL C R

U 0,−1 2,−3 1, 1M 2, 4 −1, 1 2, 2

A D 1, 2 0, 2 1, 4

So no strategies are dominant for either player. But some strategies arecertainly dominated. Maybe we can simplify the game by removing thosestrategies?

Example

BL C R

U 0,−1 2,−3 1, 1M 2, 4 −1, 1 2, 2

A D 1, 2 0, 2 1, 4

So no strategies are dominant for either player. But some strategies arecertainly dominated. Maybe we can simplify the game by removing thosestrategies?

Iterated Deletion of Dominated Strategies

Step 1: For each player, eliminate all of his dominated strategies.

Step 2: If you deleted any strategies during Step 1, repeat Step 1.Otherwise, stop.

If the process eliminates all but one strategy profile s∗, we call it adominant strategy equilibrium or we say it is the outcome of iterateddeletion of dominated strategies. Think of it as a “group process ofelimination”.

Example

Suppose it is first and ten. What should offense and defense shouldfootball teams use?

DefenseDefend Run Defend Pass Blitz

Run 3,−3 7,−7 15,−15Offense Pass 9,−9 8,−8 10,−10

(If you really like football, think of these numbers as the average number ofyards for the whole drive, given a particular strategy profile chosen above.)

Example

This can even work for large, complicated games.

a b c d eA 63,−1 28,−1 −2, 0 −2, 45 −3, 19B 32, 1 2, 2 2, 5 33, 0 2, 3C 54, 1 95,−1 0, 2 4,−1 0, 4D 1,−33 −3, 43 −1, 39 1,−12 −1, 17E −22, 0 1,−13 −1, 88 −2,−57 −2, 72

Quantity Competition

Suppose there are two firms, a and b, who choose to produce quantities qaand qb of their common product. Each firm can choose to product either1, 2, or 3 units. They have no costs, and the market price isp(qa, qb) = 6− qa − qb. The firm’s payoffs, then, are

πA(qa, qb) = p(qa, qb)qa = (6− qa − qb)qa

andπB(qb, qa) = p(qa, qb)qb = (6− qa − qb)qb

Does the game have a dominant strategy equilibrium?

Quantity Competition

Then we get a strategic form:

1 2 31 4, 4 3, 6 2, 62 6, 3 4, 4 2, 33 6, 2 3, 2 0, 0

Many games aren’t dominance solvable

But recall the Battle of the Sexes game:

l ru 2, 1 0, 0d 0, 0 1, 2

This game isn’t dominance solvable. What do we do now?

Pure-Strategy Nash Equilibrium

Definition

A strategy profile a∗ = (a∗1, a∗2, ..., a

∗n) is a pure-strategy Nash equilibrium

(PSNE) if, for every player i and any other strategy a′i that player i couldchoose,

ui (a∗i , a∗−i ) ≥ ui (a

′i , a∗−i )

A strategy profile is a Nash equilibrium if all players are using a“mutual-best response”, or no player can change what he is doing and geta strictly higher payoff. Notice that we’re thinking about the structure ofthe game, and not the motivations of any individual player, as withstrategy dominance.

Definition

ui (a∗i , a∗−i ) ≥ ui (a

′i , a∗−i )

A strategy profile is a Nash equilibrium if all players are using a“mutual-best response”, or no player can change what he is doing and geta strictly higher payoff.

Notice that we’re thinking about the structure ofthe game, and not the motivations of any individual player, as withstrategy dominance.

Definition

ui (a∗i , a∗−i ) ≥ ui (a

′i , a∗−i )

A strategy profile is a Nash equilibrium if all players are using a“mutual-best response”, or no player can change what he is doing and geta strictly higher payoff. Notice that we’re thinking about the structure ofthe game, and not the motivations of any individual player, as withstrategy dominance.

How to find PSNE’s in Strategic Form Games

Finding Nash equilibria in strategic form can done quickly:

Pick a row. Underline the best payoff the column player can receive.Check all rows.

Pick a column. Underline the best payoff the row player can receive.Check all columns.

If any box has both pay-offs underlined, it is a pure-strategy Nashequilibrium.

Example

Consider the following game:

A U 2, 1 1, 0D 1,−1 3, 3

Nash Equilibria in our Classic Games

Prisoners’ Dilemma:

s cs −3,−3 −7,−1c −1,−7 −5,−5

So our new tool — PSNE — agrees with our prediction from IDDS.

Prisoners’ Dilemma:

s cs −3,−3 −7,−1c −1,−7 −5,−5

So our new tool — PSNE — agrees with our prediction from IDDS.

Battle of the Sexes:

F BF 2, 1 0, 0B 0, 0 1, 2

There are two PSNE: (F ,F ) and (B,B). So PSNE can make usefulpredictions where dominance solvability does not.

Battle of the Sexes:

F BF 2, 1 0, 0B 0, 0 1, 2

There are two PSNE: (F ,F ) and (B,B). So PSNE can make usefulpredictions where dominance solvability does not.

Guess Half the Average

At the county fair, a farmer proposes the following game: Thetownspeople all guess the weight of a large pumpkin pie, and the personwho is closest to half the average of the guesses gets her guess in poundsof pumpkin pie, and no one else gets anything. No one is quite sure howlarge the pie is, but they all have an estimate.

More formally,

Each townsperson i = 1, 2, ...,N has a best estimate wi of the pie’sweight. They each get to submit a guess gi > 0.

The average guess is

N(g1 + g2 + ...+ gN)

The townsperson with the guess gi closest to g gets a payoff of gi .Everyone else gets nothing

What is the pure-strategy Nash equilibrium of the game?

Guess Half the Average

At the county fair, a farmer proposes the following game: Thetownspeople all guess the weight of a large pumpkin pie, and the personwho is closest to half the average of the guesses gets her guess in poundsof pumpkin pie, and no one else gets anything. No one is quite sure howlarge the pie is, but they all have an estimate. More formally,

Each townsperson i = 1, 2, ...,N has a best estimate wi of the pie’sweight. They each get to submit a guess gi > 0.

The average guess is

N(g1 + g2 + ...+ gN)

The townsperson with the guess gi closest to g gets a payoff of gi .Everyone else gets nothing

What is the pure-strategy Nash equilibrium of the game?

Hotelling’s Main Street Game

Suppose there are customers uniformly distributed along Main Street,which is one mile long. Then on the interval [0, 1], whenever1 ≥ b ≥ a ≥ 0, there are b − a customers in the subinterval [a, b]. Thereare two gas stations, a and b trying to decide where to locate their gasstations in [0, 1]; call these locations xa and xb. All customers visit theclosest gas station, and buy an amount of gasoline that gives the gasstation profits of 1 per customer. Do the players have weakly dominantstrategies? What are the Nash equilibria of the game?

Rock-Paper-Scissors:

R P SR 0, 0 −1, 1 1,−1P 1,−1 0, 0 −1, 1S −1, 1 1,−1 0, 0

And we have at least one “class” of games that don’t have pure-strategyNash equilibria: No strategy profile is underlined twice, so there are nopure-strategy Nash equilibria.

Rock-Paper-Scissors:

R P SR 0, 0 −1, 1 1,−1P 1,−1 0, 0 −1, 1S −1, 1 1,−1 0, 0

And we have at least one “class” of games that don’t have pure-strategyNash equilibria: No strategy profile is underlined twice, so there are nopure-strategy Nash equilibria.

Mixed-strategy Nash equilibria

For games like RPS, where there is no pure-strategy Nash equilibrium,there will exist a “mixed-strategy” Nash equilibrium where playersbehave randomly

We won’t cover it, since we won’t want people to behave randomly inour markets (exception: auditing)

Some games with PSNE also have MSNE: Battle of the sexes

Nash is important for proving this:

Theorem

In any game with a finite number of players and pure strategies, a(mixed-strategy) Nash equilibrium is guaranteed to exist.

Theorem

Dominant Strategy, Pure-Strategy, and Mixed-StrategyNash Equilibria

If a game is solvable by iterated deletion of dominated strategies, theoutcome is a pure-strategy Nash equilibrium, but not all pure-strategyNash equilibria are the result of iterated deletion of dominatedstrategies

For any game with a finite number of players and strategies, amixed-Nash equilibrium exists. (This result is what Nash won theNobel prize for.)

How to interpret Nash Equilibria

The Outcome of Strategic Reasoning: The logical end result of eachplayer trying to reason about what their opponents will do, knowingthe others are doing the same thing.

Norms and Conventions: The strategies that can be predicted asstable “norms” or “conventions” in society, where — given that aparticular norm has been adopted — no single person can change theconvention.

The Outcome of “Survival of the Fittest”: Suppose we have a largepopulation of players, and those who get low payoffs are removedfrom the game, while those who get high payoffs remain. As thisgame evolves, the stable outcomes of the dynamic process are Nashequilibria. (This is one foundation for evolutionary biology.)

Criticisms of Nash and Dominant Strategy Equilibria

Multiplicity of Equilibria: If a game has multiple equilibria, how dothe players know which one to use?

Computability of Equilibria: In very large or complicated games, howcan players do IDDS or find pure-strategy Nash equilibria?

Plausibility of Equilibria: In practice, many people don’t confess inprisoners’ dilemma games.

Quick Preview: Mechanism design

Mechanism design is the “inverse” of game theory

Game theory asks, given a game {Ai , ui (a)}Ni=1, what are reasonablepredictions as to what the players will do?

Mechanism design asks, given the players’ preferences ui (x) overoutcomes x , {ui (x)}Ni=1, what games cause a particular outcome x∗

to arise as a (dominant strategy or Nash) equilibrium? How do wedesign the actions for the players to induce them to choose actionsthat result in x∗?

If a game exists which induces the players to select actions leading tox∗ in a (dominant strategy or Nash) equilibrium, we say the gameimplements x∗.

game theory - university of notre dametjohns20/marketdesign_fa16/gametheory.slide… · game theory...

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