Introduction to Game Theory

Topics Dominant Strategy Nash Equilibrium Sequential Games and Subgame Perfection Backwards Induction Mixed Strategy Multi-period Games and Reputation Asymmetric Information Problems

What are in a game? Number of players Objectives of the players (payoffs) Rules Order of Play (simultaneous, sequential) Single period or multi-periods Information (common knowledge, private information)

Two companies, New product Total market: 20,000 units. Each unit sells for $10. If both enter the market, each sells half of the total -- 10,000 units.

Company A Fixed cost $60,000 Costs $5 per unit If company B doesn’t enter market, profit is $40,000. If company B enters market, profit is -$10,000. Two choices: to enter market or not to enter market.

Company B It has two technologies, old and new. The old technology has a fixed cost $30,000 and costs $6 per unit. The new technology has a fixed cost $80,000 and costs $3 per unit. If company A doesn’t enter market, profit is $50,000 and $60,000 for the two technologies, respectively.. If company A enters market, profit is $10,000 and -$10,000 for the two technologies, respectively. Three choices: to enter market with old technology, with new technology or not to enter market.

company B enter with old enter with new not enter enter -10, 10 -10, -10 40, 0 0, 50 0, 60 not enter 0, 0

company B enter with old enter with new not enter enter -10, 10 -10, -10 40, 0 0, 50 0, 60 not enter 0, 0

Dominance and iterated dominance A strategy dominates other strategies if it is a better one no matter what the other player does. A strategy is successive dominant if it is dominant after some of the opponent’s strategies are ruled out by dominance.

An important assumtion Every player is rational and “smart”. Every player knows that every player is rational and “smart”. Every player knows that every player knows that every player is rational and “smart”. .......

Nash Equilibrium Two investors to invest in a project. If both of them invest, it will succeed and generate $10K for each of them. If only one invests, the project will fail and he will lose his investment. If none invests, nothing happens.

Nash Equilibrium invest not invest invest 10, 10 -10, 0 0, 0 0, -10

Nash Equilibrium John F. Nash, 1994 Nobel Prize for “Pioneering analysis of equilibria in the theory of non-cooperative games.” A Nash Equilibrium is a set of strategies, one strategy for each player such that no player has an incentive to unilaterally change his strategy.

Sequential Games In a sequential game, players move sequentially.

Labor Contract A employee in a contract can decides whether he will follow the contract or break it and try to re-negociate If he follows the contract, he will be paid $1 and the company will get $3 If he re-negociates and the company refuses, he will get $0 and the company will get $1 because the cost of traing new employee If he re-negociates and the company agrees, he will get $2 and the company will get $2.

Labor Contract A Nash equilibrum: the employee continues the old contract and the company refuses to re-negociates. But there is a problem ...

Labor Contract (2, 2) employee company agree re-negociate refuse continue (0, 1) (1, 3)

Sub-game Perfection A subgame perfect Nash equilibrium is a Nash equilibrium for the game and, moreover, it is a Nash equilibrium for every subgame.

Backwards Induction A method of finding subgame perfect equilibrium by solving backwards of the end of the game

Backwards Induction A project has two parts. Two partners, each is in charge of one part. Each part costs $20K and will have 50% return on the investment. The first partner decides whether to invest first. If he decides to invest, (he needs to put in $20K), he will finish the one part of the project first. The project is worth $30K after the first part is finished. After that, the second partner can decide whether or not to finish the second part. If not, he will sell the unfinished project and take half of the money and leave. By doing so, his profit is $15K and leaves $15K for the first partner. Deduct the investment, the first partner will be left with a profit of -$5K. Or he can invest another $20K and finish the project. The finished project will be worth $60. Two partners share the profit, each gets $10K as profit after deducting the investment.

Backwards Induction partner 1 partner 2 (10, 10) Invest Invest not not (0, 0) (-5, 15)

Odd-Even Game 1 finger 2 finger 1 finger (0, 1) (1, 0) (0, 1) 2 finger

Mixed Strategy The Nash equilibrium for the odd-even game is that both players will mix their strategies. That is they will use 1 finger with 50% probability and 2 finger with 50% probability.

Sales Offer Goldman saleman offers a security for $5 He claims that it is worth $6. They are willing give you a good deal because you are such a good customer. He may be telling truth or trying to cheat. If he is honest, he will made $2 from the trade. If he cheats, the security is only worth $4 he will make $3 from the trade. If he is honest but you decline the offer, you will get worse deal next time and lose $1, he will make additional $1 next time.

Sales Offer honest dishonest accept (1, 2) (-1, 3) (0, 0) decline (-1, 1)

Mixed Strategy Let’s find a Nash equilibrium You will accept the offer with probability x such that he will be indifferent between being honest and dishonest. He will be honest with probability y such that you will be indifferent between accepting and declining. 3x+0(1-x)=2x+1(1-x) ===> x=1/2 y+(-1)(1-y)=(-1)y+0(1-y) ===> y=1/3 Under this equilibrium, you will lose 1/3 and he will make 3/2. Note everyone is better off if he is honest and you trade but it is not a Nash equilibrium.

To Coorperate or Not to Coorporate This is a real game played on TV each day. Two contestants win a game and can share the price of $20000. But before they take the money, each of them has to press one of the two buttons, marked “friend” and “foe”. Each contestant can’t see what the other contestant presses. If both press “friend”, they will split the money. If both press “ foe”, they will get nothing. If one presses “friend” and the other presses “foe”, the one who presses “foe” gets all the money.

To Coorperate or Not to Coorporate friend foe friend (10, 10) (0, 20) (0, 0) foe (20, 0)

To Coorperate or Not to Coorporate The only Nash equilibrium is that both press “foe”.

Multi-period Game If the game if played infinite times, the players will coorporate for fear of punishment by the other player in later rounds. One Nash equilibrium: Player 1’s strategy: player 1 presses “friend” as long as player 2 does the same. If player 2 presses “foe” once, player 1 will press “foe” forever. Player 2’s strategy is similar.

Coorporating in Competition Two firms produce and sell similar product. It cost $2 to produce 1 unit. It can sell for either $3.5 or $4. The total market will buy 10000 units. If both firms charge the same price, they spli the market. Otherwise, the whole market goes to the lower price.

Coorporating in Competition charge $3.5 charge $4 charge $3.5 (7.5, 7.5) (15, 0) (10, 10) charge $4 (0, 15)

Another way to keep Coorporating Giving rebate to match prices.

Asymmetric Information There are two types of used cars for sell: peaches and lemons. A peach is worth $3000 to a buyer and $2500 to a seller. A lemon is worth $2000 to a buyer and $1000 to a seller. There are twice as many lemons as peaches. There are much more buyers than sellers, so it is a seller’s market. The seller knows whether it is a lemon but the buyer doesn’t know.

Used Car Sales If price is below $2500, no peach will be sold. If price is above $2500, the car has a 2/3 chance to be a lemon, so it is worth $2333.33 to the buyer. So, the buyer will only pay up to $2333.33 but this won’t be enough for the seller of a peach. Conclusion: only lemons are sold at $2000.

Used Car Sales If there are twice as peaches as lemons, the buyer is willing to pay $2666.66, this is enough for all the sellers. The equilibrium is that every car sells for 2666.66 This is bad for the sellers of peaches but good for the sellers of lemons. Incentive to have a good signal. (For example, offer warrantee.)

Updating Probabilities A player’s action may reveal his information. This should be consider when choosing a strategy.

Winner’s Curse A seller trys to sell an piece of art in an auction. A buyer is bidding for it. The art is worth between $0 and $1 for the buyer, uniformly distributed. The art is worth 150% more for the buyer than for the seller. The seller knows what the art is worth but the buyer doesn’t know. Would it be a good strategy for the buyer to bid something no more than $0.5?

Winner’s Curse The buyer bids $x. If the seller doesn’t agree to sell, the buyer bidded too low and won’t get it. If the seller agrees to sell, it must be worth no more than $x for him. So it is worth no more than $1.5x for the buyer. On average, it is worth $0.75x for the buyer, bad deal. Conclusion: the winner of the auction can not win.

Problems

Backwards Induction A project has 20 parts. Two partners, each is in charge of one part. Each part costs $20K and will have 50% return on the investment. The first partner decides whether to invest first. If he decides to invest, (he needs to put in $20K), he will finish the one part of the project first. The project is worth $30K after the first part is finished. After that, the second partner can decide whether or not to finish the second part. If not, he will sell the unfinished project and take half of the money and leave. By doing so, his profit is $15K and leave $15K for the first partner. Deduct the investment, the first partner will be left with a profit of -$5K. Or he can invest another $20K and finish the project. The finished project will be worth $60.

Backwards Induction After that, the first partner can decide whether or not to finish the third part. If not, he will sell the unfinished project and take half of the money and leave. By doing so, he will get $30K and leave $30K for the second partner. Deduct the investment, the both partners will be left with a profit of $10K. Or he can invest another $20K and finish the third part of the project. The finished project will be worth $90. ........ If all 20 parts are finished, the two partners will share the profit, both of them will be left with a profit of $100 after deducting their investments.

1 2 1 1 2 1 2 ...... 100 -5 15 10 80 75 95 90 85 105

Which tire is broken? LF RF LB RB LF 1, 1 0, 0 0, 0 0, 0 RF 0, 0 1, 1

Problems Too many equilibria. How to get to an equilibrium. Focal point. Learning.