Game Theory Game theory models strategic behavior by agents who understand that their actions affect the actions of other agents. Been used to study –Market competition – Cournot Oligopoly. –Military strategies. –Bargaining/Negotiations –Biology A game consists of players, strategies, and payoffs.

Battle of Bismarck Sea Imamura Kenny 1 North South North South Imamura wants to transport troops. Kenny wants to bomb Japanese troops.. North route is two days, Southern route is three days. It takes one day for Kenny to switch routes.

Imamura wants to run convoy from Rabaul to Lae

A market analogue? Two companies, K and I, trying to maximise their shares of a market of constant size by choosing between two product designs N and S. N is similar to I’s old design and would keep some of its customers even head-to-head with K. I is a much smaller company and can only make one design. K is much larger company and can afford to start with one design and switch. K has a marketing advantage, and would like to compete head-to-head with I, while I would rather carve out its own niche instead of head-to-head competition.

Imamura Corp. 60 N S N S 70 N S Time 1Time 2 65 N S N S Time 1+2 Kenny

The Prisoners’ Dilemma Clyde Bonnie C S C S Bonnie and Clyde are caught. They can confess or be silent.

War Mars Venus Shoot Not Shoot Shoot Not Shoot Are we doomed to the bad outcome? Not in trench warfare of WWI. This happens since the game is repeated. Tit-for-tat can work. Not so easy if uncertainty of action.

The Capacity Game GM Ford 20 Expand DNE Expand DNE What is the equilibrium here? Where would the companies like to be?

Information Technology Problem Phones, Faxes, , etc. all have the following property: –Network externalities: The more people using it the more benefit it is to each user. Computers, VCRs, PS2s, also have this property in that both software can be traded among users and the larger the user market, the larger number of software titles are made. How do markets operate with such externalities?

Discussion points Competitors: Sony vs. Beta, Qwerty vs. Dvorak, Windows vs. Mac, Playstation vs. Xbox. Does the best always win? Standardization helps with network externalities. –Drive on left side vs. right side. Out of 206 countries 144 (70%) are rhs. –Left is more nature for an army: swords in right hand, mounting horses. (Napolean liked the other way.) –Sweden switched from left to right in Lots of networks: Religions and Languages.

Coordination Problem Jim Sean 0.5 Beta VHS Beta VHS Jim and Sean want to have the same VCR. Beta is a better technology than VHS.

Nash equilibrium A Nash equilibrium is a set of strategies – Where each player has no incentive to deviate. –Given other equilibrium strategies a player would choose his equilibrium strategy. –A best response to a best response. A pure strategy equilibrium is where each player only chooses a particular strategy with certainty. What are the pure-strategy equilibria in Prisoners’ dilemma and the coordination game. Is there always a pure strategy equilibrium?

Penalty Kick Goalie Kicker 1 Kick R Dive L Dive R Kick L A Kicker can kick a ball left or right. A Goalie can dive left or right.

Mixed Strategy equilibrium Happens in the Penalty kick game. Notice that if the Kicker kicks (.5L+.5R), the Goalie is indifferent to diving left or right. If the Goalie dives (.5L+.5R), the Kicker is indifferent to kicking left or right. Thus, (.5L+.5R,.5L+.5R) is a mixed-strategy N.E. Nash showed that there always exists a Nash equilibrium.

Do you believe it? Do they really choose only L or R? Yes. Kickers 93.8% and Goalie 98.9%. Kickers are either left or right footed. Assume R means kick in “easier” direction. Below is percentage of scoring. Nash prediction for (Kicker, Goalie)=(41.99L+58.01R, 38.54L+61.46R) Actual Data =(42.31L+57.69R, 39.98L+60.02R) Kick R Dive L Dive R Kick L

Parking Enforcement Game Student Driver University 5 Don’t Park OK Park in Staff Check Student can decide to park in staff parking. University can check cars in staff parking lot.

What happens? If the University checks, what do the students do? If the students park ok, what does the Uni do? If the uni doesn’t check, what do the students do? If the students park in the staff parking, what does the uni do? What is the equilibrium of the game? What happens if the university makes it less harsh a punishment to only –10. Who benefits from this? Who is hurt by this?

Answer Student parks legally 1/3 of the time and the uni checks 1/10 of the time. With lower penalty, student parks legally 1/3 of the time and the uni checks 2/3 of the time. Who’s expected payoff changes? No one.