chapter 4: games and strategy

Definition of a GameDominant Strategies and Nash Equilibrium

Sequential Games

Chapter 4: Games and Strategy

Kerry Tan

June 25, 2009

Economics 570: Government and Business Games and Strategy


Sequential Games

What is a Game?

game

A stylized model that depicts situations of strategic behavior,where the payoff for one agent depends on its own actions aswell as on the actions of the other agents.

A game consists of:playersa set of rules (actions for a player)payoff (what is the utility for a player from a particularoutcome)

Ultimate goal: Predict the outcome of the game.



Sequential Games

Prisoner’s Dilemma

Two suspects in a crime are put into separate cells. If they bothconfess, each will be sentenced to three years in prison. If onlyone of them confesses, he will be freed and used as a witnessagainst the other, who will receive a sentence of four years. Ifneither confesses, they will both be convicted of a minoroffense and spend one year in prison.

How do we model this type of scenario?Normal form representation: Use a matrix where each cellcorresponds to a combination of strategic choices by eachplayer.



Sequential Games

Dominant StrategiesNash Equilibrium

Dominant Strategies

How do we solve the Prisoner’s Dilemma game?We first need to look for the dominant strategy for each player.

dominant strategyWhenever a player has a strategy that is strictly better than anyother strategy regardless of the other players’ strategy choices.

If a player has a dominant strategy and if the player is rational,then we should expect the player to choose the dominantstrategy.

What is the dominant strategy for each player in the Prisoner’sDilemma game?Each suspect would choose to confess.



Sequential Games


Battle of the Sexes Game

Does every game have a dominant strategy? No.

Suppose a boyfriend and girlfriend are deciding about what todo on a date. Their two options are to either go to COSI or tothe Columbus Zoo. Each person would rather do somethingtogether than not do something together. The guy gets 2 utilsand the girl gets 1 util if they end up going to COSI. The guygets 1 util and the girl gets 2 utils if they end up going to thezoo. They get zero utils otherwise.

No dominant strategy for either person in this case.



Sequential Games


Nash Equilibrium

Nash EquilibriumA pair of strategies constitutes a Nash equilibrium if no playercan unilaterally change its strategy in a way that improve itspayoff.

In other words, given the other players’ strategies, no playercan profitably deviate.

Nash equilibrium for Prisoner’s Dilemma game:(Confess, Confess)Nash equilibrium for Battle of the Sexes game:(Mall, Mall) and (Crew Game, Crew Game)



Sequential Games


Matching Pennies Game

Does every game have a Nash equilibrium? No.

Each player has a penny and must secretly turn the penny toheads or tails. The players then reveal their choicessimultaneously. If the pennies match (both heads or both tails),Player 1 receives one dollar from Player 2. If the pennies do notmatch (one heads and one tails), Player 2 receives one dollarfrom Player 1.

In this case, there is no Nash equilibrium.



Sequential GamesSubgame Perfect Equilibrium

Simultaneous Movie Release Game

Suppose Warner Brothers is trying to figure out when to set therelease date for Terminator Salvation. At the same time, Sonyis determining when to release Angels & Demons. Both want torelease the movie around Memorial Day due to the spike indemand for movies. If they both release their movies on May 21(Memorial Day weekend), then they split the huge demand andboth movies will make $50 million that weekend. If they bothrelease their movies on May 15 (the week before MemorialDay), then both movies will make $30 million. If one moviereleases on May 21 and the other on May 15, then the moviethat ends up releasing on May 21 will reap the rewards of thehigh demand and make $90 million, whereas the moviereleasing on May 15 will make $55 million.

Nash equilibria: (May 15, May 21) and (May 21, May 15)Economics 570: Government and Business Games and Strategy



Sequential Games

So far we have only considered simultaneous-choice games.But what happens when choices are made sequentially byplayers?

sequential movie release game: Suppose Warner Brothersdecides on its release date for Terminator Salvation first. Uponseeing this, Sony is able to decide its release date for Angels &Demons. What do we expect to happen?

How do we model this type of scenario?Extensive form representation: Use a game tree.




Extensive Form Representation

A game tree is like a decision tree except that there is morethan one decision maker involved.

A game tree consists of:decision node: The point that specifies which player ismaking an action.branch: the strategies a player can make from a specificdecision node.terminal node: the point where the game ends and utilityfor each player is specified.




Subgame

A subgame in an extensive-form game with perfect informationbegins at a decision node (but is not the game’s firstdecision node).includes all the decision and terminal nodes following thatdecision node in the game tree.

There is one subgame in the sequential entry game.




Backward Induction

We use backward induction in order to get rid of “unreasonable"Nash equilibrium strategies.

Consider the last node of the game.Determine what the optimal decision would be.Solve for the decision node above given the decisionpreviously found.Repeat this process until you reach the first decision node.




Sequential Movie Release Game

First take a look at the decision node for Sony.Suppose Sony thinks that Warner Brothers will releaseTerminator Salvation on May 15.

If Sony chooses to release Angels & Demons on May 15,they will get a profit of $30 million.If Sony chooses to release Angels & Demons on May 21,they will get a profit of $90 million.Since 90>30, then Sony would choose to release Angels &Demons on May 21.

Suppose Sony thinks that Warner Brothers will releaseTerminator Salvation on May 21.

If Sony chooses to release Angels & Demons on May 15,they will get a profit of $55 million.If Sony chooses to release Angels & Demons on May 19,they will get a profit of $50 million.Since 55>50, then Sony would choose to release Angels &Demons on May 15.





Next take a look at the decision node for Warner Brothers.If Warner Brothers chooses to release Terminator Salvationon May 15, they expect that Sony will choose to releaseAngels & Demons on May 21.

Warner Brothers would get a profit of $55 million.If Warner Brothers chooses to release Terminator Salvationon May 21, they expect that Sony will choose to releaseAngels & Demons on May 15.

Warner Brothers would get a profit of $90 million.

Since 90>55, then Warner Brothers would choose torelease Terminator Salvation on May 21.




Subgame-Perfect Equlibrium

Subgame-perfect equilibrium: An equilibrium strategy derivedfrom backward induction.

Subgame-perfect equilibrium is a subset of Nashequilibrium.It eliminates “unreasonable" Nash equilibrium strategies.Subgame-perfect equilibrium consists of credible Nashequilibrium strategies.





The subgame-perfect equilibrium is (May 21, May 15).Note: There is only one equilibrium in the sequential game,but there were two in the simultaneous version of thegame.Warner Brothers got the advantage from moving first.

This is often called the first-mover advantage.




Entry Game

Consider the example of an industry that is currentlymonopolized. A second firm must decide whether to enter ornot to enter the industry. The incumbent observes whether ornot the entrant enters, and then decides whether or not to priceaggressively. If the entrant does not enter, then the entrantmakes a profit of $0 and the incumbent makes a profit of $50. Ifthe entrant enters and the incumbent retaliates, then both theentrant and incumbent make -$10. If the entrant enters and theincumbent does not retaliate, then the entrant makes $10 andthe incumbent makes $20.




Normal Form Representation

There are two Nash equilibrium strategies:(Don’t Enter, Retaliate) and (Enter, Don’t Retaliate)

Does (Don’t Enter, Retaliate) make sense?Player 1 is not entering because of the "threat" that Player2 will choose to retaliate.This threat is not credible, since Player 2 would notretaliate if Player 1 were to enter.By retaliating, Player 2 gets -10, compared with 20 from noretaliation.

Although (Don’t Enter, Retaliate) is a Nash equilibrium, it is nota reasonable prediction of what one might expect to be played.




Backward Induction in the Sequential Entry Game

First take a look at the decision node for the monopolist.If the monopolist chooses retaliate, it would get a profit of-10.If the monopolist chooses don’t retaliate, it would get aprofit of 20.Since 20>-10, then the monopolist would chose don’tretaliate.




Backward Induction in the Sequential Entry Game

Given that the monopolist would rationally choose to notretaliate, then determine what the entrant would do.

If the entrant enters, then it would get a profit of 10 (as theentrant knows the monopolist would choose to not retaliatein this case)If the entrant doesn’t enter, then it would get a profit of 0.Since 10>0, the entrant would choose to enter.

Subgame-perfect equilibrium of sequential-entry game:(Enter, Don’t Retaliate)


chapter 4: games and strategy

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