Microeconomics 4. Game Theory - newes.saif.sjtu.edu.cn

Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory

Microeconomics4. Game Theory

Jun Li

Shanghai Advanced Institute of Finance

Microeconomics Jun E. Li SAIF 1 / 60


Overview

1 Introduction

2 Nash EquilibriumWeakly Dominant StrategiesNash Equilibrium

3 Sequantial Games and Repeated GamesSequential GamesRepeated Games

4 Mixed StrategiesMixed Strategy Equilibria

5 More ApplicationsGames of CoordinationGames of Competition (Zero-Sum Games)Games of Commitment

6 Summary of Game theorySummary



Introduction

Game theory is a set of analytical tools designed to understand theinteractions of decision makers.

What is a game? Super Mario, Counter Strike, Star Craft, DOTA,LOL, WOW, PUBG?

”A game is a description of strategic interaction that includes theconstraints on the actions that the players can take and the players’interests, but does not specify the actions that the players do take.”— Osborne and Rubinstein (1994)

What is a solution to a game? ”A systematic description of theoutcomes that may emerge in a family of games.”

Game theory suggests reasonable solutions for games.

Foundation for general equilibrium theory.



Introduction

Game theory is a set of analytical tools designed to understand theinteractions of decision makers.

What is a game? Super Mario, Counter Strike, Star Craft, DOTA,LOL, WOW, PUBG?”A game is a description of strategic interaction that includes theconstraints on the actions that the players can take and the players’interests, but does not specify the actions that the players do take.”— Osborne and Rubinstein (1994)

What is a solution to a game? ”A systematic description of theoutcomes that may emerge in a family of games.”

Game theory suggests reasonable solutions for games.

Foundation for general equilibrium theory.



Basic assumptions

What is a strategic game consist of?

N fully rational agents: agent i is aware of alternatives, formexpectations of unknowns, chooses actions which maximize theutility/payoff

For each agent i that he/she can choose from strategy space Si

agent i’s strategy given all others’ strategy: (si, s−i)

si: agent i’s strategy

s−i: all others’ strategy

Each agent i’s payoff/utility are specified, which indicate theutility/payoffs once a certain combinations of strategies is chosen

utility u(si, s−i) describes agent i’s utility given his strategy si andall others’ strategy s−i

In this lecture all agents have complete information

Knowing other players’ possible strategies, probabilities, objectives...



Matching pennies game in matrix form

B (Columns)Heads Tails

A (Rows)Heads 1, -1 -1, 1

Tails -1, 1 1, -1

Both players secretly turn a penny to heads or tails.

They reveal their penny simultaneously.

If the pennies match, A collects them, otherwise B collects them.

(Heads, Tails) in the upper right indicates A gets -1 and B gets 1

The sum of the payoffs of A and B is zero, this type of game is alsocalled zero-sum game



Table of Contents

1 Introduction








Strategies

There are lots of strategies you may know from the Chinese history

Thirty-six stratagems (5th century BC)

3. Kill with a borrowed knife

15. Lure the tiger off its mountain lair

32. Empty fort strategy (The Three Kingdoms)

36. Run away (the most well known one)

Tian Ji’s horse racing strategy



Weakly Dominant Strategy

Weakly Dominant Strategy

A strategy s∗i is a weakly dominant strategy if it is better than all otherstrategies regardless of the opponents’ strategies,

ui(s∗i , s−i) ≥ ui(s′i, s−i)

for all s−i ∈ S−i and all s′i ∈ Si, s′i 6= s∗i

Strategy s∗i weakly dominates s′i if the payoff from strategy s∗i isweakly preferred than the payoff for s′i no matter what choices theother players take.

s∗i does not need to be unique, since ≥ is imposed

s∗i may not exist

s∗i is among the best strategies



Weakly Dominated Strategy

Weakly Dominated Strategy

A strategy s′i is weakly dominated if there exist some si ∈ Si such that

ui(si, s−i) ≥ ui(s′i, s−i)

for all s−i ∈ S−i.

The strategies that are dominated

Being dominated by one other strategy is sufficient



Strictly Dominant and Dominated Strategy

Strictly Dominant Strategy

A strategy s∗i is a strictly dominant strategy if it is better than all otherstrategies regardless of the opponents’ strategies,

ui(s∗i , s−i) > ui(s

′i, s−i)

for all s−i ∈ S−i and all s′i ∈ Si, s′i 6= s∗i

Strictly Dominated Strategy

A strategy s′i is strictly dominated if there exist some si ∈ Si such that

ui(si, s−i) > ui(s′i, s−i)

for all s−i ∈ S−i. A strictly dominated strategy is never optimal thusnever been chosen.



Elimination of strictly dominated strategies



Another example



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1 Introduction








Nash Equilibrium

Nash Equilibrium

A Nash Equilibrium of a strategic game is a profile s∗ ∈ S of strategieswith the property that for every agent i we have

u(s∗i , s∗−i) ≥ u(s′i, s

∗−i)

for all s′i ∈ Si for all agents i. No player wants to deviate.

Difference between Nash Equilibrium and equilibrium in dominantstrategies

N.E.: each strategy is best response to others’ equilibrium strategies

Dominant strategy: optimal no matter what the other does

Equilibria in dominant strategies are Nash equilibria

But not all N.E. are equilibria in dominant strategies



Iterated elimination of dominated strategies may not work

Left Middle Right

Top 0, 4 4, 0 5, 3

Middle 4, 0 0, 4 5, 3

Bottom 3, 5 3, 5 6, 6

No dominated strategies for player 1, neither for player 2.

Nash equilibrium still exists.



Two Nash Equilibria?

BBQ Salad

BBQ 2, 2 0, 2

Salad 2, 0 1, 1

(BBQ, BBQ) and (Salad, Salad) are two N.E. for this game

Will they reach agreement on having salad for dinner?

Is it optimal for player 2 (columns) to choose BBQ?

By using the elimination of weakly dominated strategies, No

If we eliminate dominated strategies, we will eliminate some NashEquilibria



Two Nash Equilibria?

BBQ Salad

BBQ 2, 2 0, 2

Salad 2, 0 1, 1

(BBQ, BBQ) and (Salad, Salad) are two N.E. for this game

Will they reach agreement on having salad for dinner?

Is it optimal for player 2 (columns) to choose BBQ?

By using the elimination of weakly dominated strategies, No

If we eliminate dominated strategies, we will eliminate some NashEquilibria



Dominant Strategy Equilibrium and Nash Equilibrium

L R

T a, b c, d

B e, f g, h

If (T,L) to be an equilibrium with strictly dominant strategies, thefollowing conditions must be true:

a > e, c > g, b > d, f > h

If (T,L) to be a Nash Equilibrium, the following conditions must betrue:

a ≥ e, b ≥ d

A dominant strategy equilibrium needs more stringent requirement,it is always a Nash equilibrium, but not vice versa.



Review Nash Equilibrium

Nash Equilibrium

A Nash Equilibrium of a strategic game is a profile s∗ ∈ S of strategieswith the property that for every agent i we have

u(s∗i , s∗−i) ≥ u(s′i, s

∗−i)

for all s′i ∈ Si for all agents i. No player wants to deviate.

Weakly dominant is sufficient

Strictly dominated strategies are never chosen

Self-enforcing agreement, no one deviates

Nash Equilibrium is a stable social convention

Nash Equilibrium is also the microfoundation for general equilibrium,each individual is considering other’s best strategies, and makingdecisions strategically.



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1 Introduction








Sequential games

Up until now we assume players act simultaneously

What if one player act firstly, and then other player response?



Sequential games

Let’s firstly look at the Nash equilibrium for simultaneous moves

Motorola has advantages in analog tech, Sony has advantage indigital tech

Motorala acts firstly, then Sony makes choices



Extensive form



Sequential games

The way to solve sequential game is through backwards induction

1 The last step: Sony’s choices

Conditional on Motorola chooses digital, Sony will choose the one infavor of itself (digital)

Conditional on Motorola chooses analog, Sony will choose the one infavor of itself (analog)

2 The second to last step: Motorola’s choices. Motorola knows givenits own choices, the opponent (Sony) will have different strategies

In order to maximize Motorala its own payoff, it chooses analog



Solution to this game



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1 Introduction








Repeated games

Previously, games are only played once, called one-shot game

Players never cooperate, since they only play once

If you make your opponents fail or look bad, it does not matter,because you do not see each other anymore

Often the case you and your opponents are playing repeatedly

Opponents can adjust their strategy based on the history, so do you



Repeated prisoner’s dilemma: Finite games

Cooperate Confess

Cooperate 5, 5 -1100, 10

Confess 10, -1100 -1000, -1000

Finitely repeated game: N rounds

At first you may think they try to be nice to each other

Round N : they reach (Confess, Confess)

Round N − 1: since they know both will confess in round N , theywill confess this round

Round N − 2: repeat the thought process above

This is also called backwards induction



Repeated prisoner’s dilemma: Infinite games

Cooperate Confess

Cooperate 5, 5 -110, 10

Confess 10, -110 -100, -100

Infinitely repeated game: N =∞ rounds, with a moderate discountrate δ

At first you may think they will be mean to each other as N roundcase, but there is no last round

You’ll calculate your payoffs based NPV

Tit-for-tat strategy: cooperate on the first round. For every roundthereafter, if your opponent cooperates on the previous round, youalso cooperate. Otherwise do not cooperate.

The punishment strategy constitutes a Nash Equilibrium

Think about currencies and cryptocurrencies:People hold certain currency is because they believe this currencywill be used in the infinite future.

Bitcoin has value because everybody believes it will be used in thefar future



Remarks on finite and infinite repeated games

Finite or infinite horizon? It should capture how the players perceive

Infinite horizon: for each period, players believe the game willcontinue for an additional period

Finite horizon: players clearly perceive a well-defined final period

It depends on whether the last period’s payoff directly enters theplayers’ strategic considerations.

Suppose the payoff matrix of the prisoner’s dilemma is

Cooperate Confess

Cooperate 3,3 0,4

Confess 4,0 1,1

If these two prisoners’ play this game for 20 times, with payoff wespecified here, it’s better to regard it as a infinitely repeated game.

Except very close to the end of the game, they are likely to ignorethe payoffs of the final period



Remarks on finite and infinite repeated games

The rationale is always based on the discounted utility∑Tt=1 β

tu(xt), or discounted payoffs∑T

t=1 βtxt.

If the prisoners discount future heavily (e.g., β is super small), theymay not cooperate and only play Nash equilibrium

The casual observations suggest that people act cooperately whenthe horizon is distant and opportunistically when it is near, which isconsistent with finitely repeated games

The notion of infinitely repeated games cannot give us such insightsof that behavior

The discontinuity between the outcomes of finitely and infinitelyrepeated games is not appealing.



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1 Introduction








Mixed Strategies

Previously we only discussed pure strategy: it provides a completedefinition of how an agent will play a game, the choices aredeterministic

Now mixed strategies: choices are not deterministic but followingprobability distributions

There are infinite many mixed strategies

To describe mixed strategy, one has to specify both probabilities andthe choices (example latter)



Why Mixed Strategies?

Lots of games are of probabilities:

Rock, Scissors and Paper

Goal keeper and penalty shooter

Poker game

Of course video games

Because we know opponents are choosing different strategies withprobabilities, so we choose counter strategies with probabilities, andopponents know that we know they are ...



The game in matrix form

Mr. Row chooses top with probability r, and with probability (1− r)he chooses bottom

Ms. Column chooses left with probability c, and with probability(1− c) he chooses bottom



The expected payoffs

Let’s list all possible payoffs given different combinations of choices

Combination Probability Payoff to RowTop, Left rc 2Bottom, Left (1− r)c 0Top, Right r(1− c) 0Bottom, Right (1− r)(1− c) 1

Row’s payoff = 2rc+ (1− r)(1− c) = 2rc+ 1− r − c+ rc

Suppose Row increases r by ∆r:∆ Row’s payoff = 2c∆r −∆r + c∆r = (3c− 1)∆r

Column’s payoff = cr + 2(1− c)(1− r)Suppose Column increases c by ∆c:∆ Column’s payoff = r∆c + 2r∆c− 2∆c = (3r − 2)∆c



Characterize the Nash equilibria

∆ Row’s payoff = (3c− 1)∆r

If c > 1/3, Mr. Row wants to increase r ⇒ r∗ = 1

If c < 1/3, Mr. Row wants to decrease r ⇒ r∗ = 0

If c = 1/3, Mr. Row is indifferent in the interval r ∈ [0, 1]

∆ Column’s payoff = (3r − 2)∆c

If r > 2/3, Ms. Column wants to increase c⇒ c∗ = 1

If r < 2/3, Ms. Column wants to decrease c⇒ c∗ = 0

If r = 2/3, Ms. Column is indifferent in the interval c ∈ [0, 1]

The conditions above characterize the best response of the twoplayers

We can use best response curves to represent the Nash equilibria,including pure and mix strategies



Best response curve

The two curves depict the best response of row and column to eachother’s choices. The intersections of the curves are Nash equilibria.In this case there are three equilibria, two with pure strategies andone with mixed strategies.



Obtain the mix strategy equilibrium using FOC

The analysis of small changes in ∆r,∆c can be based on FOCs

Recall ∆F (x, y) = ∂F∂x ∆x+ ∂F

∂y ∆y

Row’s payoff = R(r, c) = 3rc+ 1− r − cRow’s payoff of increasing probability ∆r: ∂R(r,c)

∂r∆r = (3c− 1)∆r

Column’s payoff = C(r, c) = 3cr + 2− 2r − 2c

Col’s payoff of increasing probability ∆c: ∂C(r,c)∂c

∆r = (3r − 2)∆c

These are exactly the same analysis, we’ll have the mix strategyequilibrium (r = 2/3, c = 1/3)

However, if you want to fully characterize the solution to this game,you still need to tell the play’s best response given the opponent’sresponse, i.e., the best response curves.



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1 Introduction








Assurance games

Nash equilibria: (refrain, refrain) and (build, build)

Neither party knows which choice the other will make. Beforecommitting to refrain, each party wants some assurance that theother will refrain

A simple way of assurance is to open itself for inspection. As long asone party gives the other party sufficient evidence that it will chooserefrain, this will help the assurance that (refrain, refrain) equilibriumcould occur.



Chicken game

Bill and Jane are lazy couples, but someone has to wash the dishes.



Volunteer’s dilemma

If at least one person writes down 1 point, then everyone gets thenumber of points they write down. If no one chooses 1 point,everyone gets 0 points.

Some examples

The murder of Kitty Genovese: 38 witnesses saw or heard the attack,and that none of them called the police or came to her aid

Meerkat: One or more meerkat act as sentries while others search forfood


https://www.history.com/topics/crime/kitty-genovese

https://baike.baidu.com/tashuo/browse/content?id=e5a238268463620f7bac3780&lemmaId=7587376&fromLemmaModule=pcRight


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1 Introduction








Penalty shot in mixed strategies

Row is kicking a penalty shot and column is defending

The payoffs are twisted so that this game is not perfectly symmetric.You could think row is good at kicking a certain direction.

Notice that this is a zero-sum game:

Maximizing one’s own payoff means minimizing the other player’spayoff



Penalty shot in mixed strategies

Let’s impose probabilities:

Row: With probability p row kicks left and 1− p kicks right

Column: With probability q column defends left and 1− q defendsright



Row’s expected payoffs

Row’s expected payoffs = 50p+ 90(1− p) if Column jumps left

Row’s expected payoffs = 80p+ 20(1− p) if Column jumps right

Keep in mind the property of this zero-sum game is that when Rowmaximizing its payoffs, it is minimizing Column’s payoffs

This implies for any p, the best payoff Row can hope for is theminimum of the payoffs given by the two possible strategiesperformed by Column



Row’s strategy

The two curves show row’s expected payoff as a function of p, theprobability that he kicks to the left. Whatever p he chooses, column willtry to minimize row’s payoff. The optimal point:50p+ 90(1− p) = 80p+ 20(1− p)



Column’s expected payoffsy

Column’s expected payoffs = 50q + 80(1− q) if Column jumps left

Column’s expected payoffs = 90 + 20(1− q) if Column jumps right

Follow the same logic as Row, we could identify the point whereRow’s maximum payoff is minimized



Column’s strategy

The two lines show row’s expected payoff as a function of q, theprobability that column jumps to the left. Whatever q column chooses,row will try to maximize his own payoff. The optimal point:50q + 80(1− q) = 90q + 20(1− q)



The best response curves



In comparison to the FOC approach

We have one mixed strategy equilibrium

You could redo the whole exercise using FOC approach, you will getexactly the same results.

The intuition is that for a zero-sum game, each agent knows whenmaximizing its own payoff, it is minimizing the opponent’s payoff.



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1 Introduction








The kindly kidnapper

There are lots of kidnappings for ransom happening around theworld.

In some countries, paying ransom is illegal. The argument is that ifthe victim’s family or employers can commit not to pay the ransom,the kidnappers will have no incentives to do so.



The kindly kidnapper

Kidnappers can choose to release or kill the hostage.

If a hostage got released, he could either identify or refrain to do so.



How to achieve commitment?

It turns out, once being released, the hostage always would like toidentify the kidnappers

How could the hostage convince the kidnappers that he wouldn’trenege on his promises

The hostage has to find a way to change the payoff structure, e.g.,impose some cost of identify the kidnappers

Photos of embarrassing act

You may heard of similar stories that two kings want to ensure acontract, a way to achieve so is to keep each other’s family membersas hostages.



Savings and social security

In lots of countries, people do not save much because they recognizethat the society would not let them starve



Savings and social security

However, old chooses firstly, it will result that old chooses squanderand young chooses support



Pay as you go

Pay as you go system is popular in lots of countries, such asEuropean countries and China.

Pay as you go system faces the situation of the game we describedbefore.

Lots of countries are running the mixed between pay as you go andthe defined benefit (401k in US). This forces each generation to savefor retirement.



Table of Contents

1 Introduction








Summary

Dominant strategy equilibrium

Nash equilibrium

Simultaneous moves

Repeated games and sequential games

Mixed strategy equilibrium

Games of coordination, competition and commitment


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Microeconomics 4. Game Theory - newes.saif.sjtu.edu.cn