microeconomics 4. game theory - newes.saif.sjtu.edu.cn
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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
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 2 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game 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.
Microeconomics Jun E. Li SAIF 3 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game 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.
Microeconomics Jun E. Li SAIF 3 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game 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...
Microeconomics Jun E. Li SAIF 4 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 5 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Table of Contents
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
Microeconomics Jun E. Li SAIF 6 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 7 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 8 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 9 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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.
Microeconomics Jun E. Li SAIF 10 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Elimination of strictly dominated strategies
Microeconomics Jun E. Li SAIF 11 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Another example
Microeconomics Jun E. Li SAIF 12 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Table of Contents
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
Microeconomics Jun E. Li SAIF 13 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 14 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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.
Microeconomics Jun E. Li SAIF 15 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 16 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 16 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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.
Microeconomics Jun E. Li SAIF 17 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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.
Microeconomics Jun E. Li SAIF 18 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Table of Contents
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
Microeconomics Jun E. Li SAIF 19 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Sequential games
Up until now we assume players act simultaneously
What if one player act firstly, and then other player response?
Microeconomics Jun E. Li SAIF 20 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 21 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Extensive form
Microeconomics Jun E. Li SAIF 22 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 23 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Solution to this game
Microeconomics Jun E. Li SAIF 24 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Table of Contents
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
Microeconomics Jun E. Li SAIF 25 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 26 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 27 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 28 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 29 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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.
Microeconomics Jun E. Li SAIF 30 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Table of Contents
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
Microeconomics Jun E. Li SAIF 31 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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)
Microeconomics Jun E. Li SAIF 32 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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 ...
Microeconomics Jun E. Li SAIF 33 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 34 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 35 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 36 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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.
Microeconomics Jun E. Li SAIF 37 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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.
Microeconomics Jun E. Li SAIF 38 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Table of Contents
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
Microeconomics Jun E. Li SAIF 39 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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.
Microeconomics Jun E. Li SAIF 40 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Chicken game
Bill and Jane are lazy couples, but someone has to wash the dishes.
Microeconomics Jun E. Li SAIF 41 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 42 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Table of Contents
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
Microeconomics Jun E. Li SAIF 43 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 44 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 45 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 46 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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)
Microeconomics Jun E. Li SAIF 47 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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
Microeconomics Jun E. Li SAIF 48 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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)
Microeconomics Jun E. Li SAIF 49 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
The best response curves
Microeconomics Jun E. Li SAIF 50 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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.
Microeconomics Jun E. Li SAIF 51 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Table of Contents
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
Microeconomics Jun E. Li SAIF 52 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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.
Microeconomics Jun E. Li SAIF 53 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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.
Microeconomics Jun E. Li SAIF 54 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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.
Microeconomics Jun E. Li SAIF 55 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Savings and social security
In lots of countries, people do not save much because they recognizethat the society would not let them starve
Microeconomics Jun E. Li SAIF 56 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Savings and social security
However, old chooses firstly, it will result that old chooses squanderand young chooses support
Microeconomics Jun E. Li SAIF 57 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
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.
Microeconomics Jun E. Li SAIF 58 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Table of Contents
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
Microeconomics Jun E. Li SAIF 59 / 60
Introduction Nash Equilibrium Sequantial Games and Repeated Games Mixed Strategies More Applications Summary of Game theory
Summary
Dominant strategy equilibrium
Nash equilibrium
Simultaneous moves
Repeated games and sequential games
Mixed strategy equilibrium
Games of coordination, competition and commitment
Microeconomics Jun E. Li SAIF 60 / 60