Game Theory II: Non-Zero-Sum Games

CSCI 5582, Fall 2007

Assignments

• Problem Set 4 due Thursday Dec. 6• Problem Set 5 due Thursday Dec. 13• Final exam Saturday Dec. 15• Review session evening of Thursday 12/13

(place and time TBA)

Evolutionary Game Theory

• Each animal can play one of two strategies:“hawk” and “dove”. Hawks always fightover a resource; doves retreat.

• When two hawks meet, they fight (assume a50 percent chance of winning).

• When two doves meet, they “display” andchoose the “winner” at random.

Red

Blue

Hawk

Dove

Hawk Dove

(50 + -100)/2 = -25

50

0 ((50 - 10) + -10)/2 = 15

Looking for an “ESS”

• Here, neither “pure hawk” nor “pure dove” is anevolutionarily stable strategy (or ESS), since each can besuccessfully invaded by the other.

• Suppose our population isn’t all one or the other, butinstead has (in percentage) P hawks and (1-P) doves. Thepayoff to a newborn hawk is -25P + 50(1-P). The payoff toa new dove is 0P + (1-P)15. When these are equal, neitherstrategy is favored. So conceivably our population couldstabilize at a percentage P = 35/60 of hawks.

Red

Blue

Hawk

Bourgeois

Dove

-25 50 12.5

0 15 7.5

-12.5 32.5 25

Hawk Dove Bourgeois

Games “in the Large”

• Games where I wish to do “X” only if fewpeople are doing “X”

• Games where I wish to do “X”only if lots ofpeople are doing “X”

• Public participation and anonymity• Behavioral game theory• I don’t want to play!

A Class-wide game

• If you choose “C”, you get N * 100 dollars.• If you choose “D”, you get 200 + (N * 100)

dollars.• N is the number of people who choose “C”.

Behavioral Game Theory

or, the “Let’s Meet in Paris” game...

I Don’t Want to Play!

• The “tragedy of the commons”• Game theoretic analysis of emotions• Ceding control:opting out of the game

200 400 600 800 1000Number of Cows

QuartsPer Cow

1000

800

600

400

200

10090,0009009009200160,00016008008300210,00021007007400240,00024006006500250,00025005005600240,00024004004700210,00021003003800160,0001600200290090,0009001001

Qts/CowTotalQuarts

MyQuarts

Total NMy N

Common sense says that victory goes to the side with themost intelligence, self-interest, coolness, options, power, andclear lines of communication. Common sense is wrong. Eachof these assets can be a liability in contests of strategy (asopposed to contests of chance, skill, or strength), wherebehavior is calculated by predicting what the other guy willdo in response....How do you persuade someone that you will not pay morethan $16,000 for a car that is really worth $20,000 to you?You make a public, enforceable $5,000 bet with a third partythat you won’t pay more than $16,000. As long as $16,000gives the dealer a profit, he has no choice but to accept....

Because an expensive threat works both ways, it can lead to acycle of self-incapacitation. Protesters attempt to block theconstruction of a nuclear power plant by lying down on therailroad tracks leading to the site. The engineer, beingreasonable, has no choice but to stop the train. The railroadcompany counters by telling the engineer to set the throttle sothat the train moves very slowly and then to jump out of thetrain and walk beside it. The protesters must scramble. Nexttime the protesters handcuff themselves to the tracks; theengineer does not dare leave the train. But the protesters mustbe certain the engineer sees them in enough time to stop. Thecompany assigns the next train to a nearsighted engineer.

- Pinker, How the Mind Works

Emotions as Game-Theoretic Strategies

Red

Blue

Red

Blue

Red

Blue