Incomplete Information and Bayes-Nash Equilibrium

Why some fights start

Weak or tough? Two players are in dispute over a resource. Player 1 knows his own strength, but does not know whether Player 2 is weak or tough and doesn’t know if Player 1 will fight or yield. If Player 2 is weak, Player 1 could beat him in a fight, but if Player 2 is tough, Player 1 would lose. Player 2 knows his own strength and also knows Player 1’s strength, but doesn’t know what Player 1 will do. Each player can either yield or fight. The extensive form of the game is shown on the next slide.

Tough or Weak? Nature 2 is strong2 is weak Player 2 Player 1 Fight Yield Top number is payoff to Player 2. Bottom is payoff to Player p1-p

What do we need to specify for a Bayes-Nash equilibrium Strategy for Player 1: Yield or Fight Strategy for Player 2: What he will do if strong. What he will do if weak.

Is there a Nash Equilbrium where Player 1 yields? If Player 1 yields, then Fight is the best response for Player 2, whether he is strong or weak. If Player 2 always plays Fight, is Yield the best response for Player 1? Expected payoff for Player 1: From Yield is 0. From Fight is -100p+100(1-p)=100(1-2p) Payoff from Yield is larger than from Fight if p>1/2

Bayes-Nash equilibrium Player 1 believes that the probability that 2 is tough is p>1/2 Player 1 Yields. Player 2 Fights if strong and Fights if weak. Expected payoff to Player 1 is 0.

When is there an equilibrium where a fight could happen? If Player 1 fights, best response for Player 2 is to Fight if he is strong and Yield if he is weak. If Player 2 fights when he is strong and yields when he is weak, expected payoff to Player 1 from Fight is -100p+100(1-p)=100(1-2p). Expected payoff to Player 1 from Yield is 0. Fight is best response if p<1/2.

Bayes-Nash Equilibrium if p<1/2 Player 1 chooses Fight Player 2 chooses Fight if he is Strong and Yield if he is weak. Expected payoff to Player 1 in this equilibrium is 100(1-2p).

Hiring a Spy How much is it worth to Player 1 to find out whether Player 2 is strong or weak? If Player 1 knows whether 2 is strong or weak, Player 1 will Yield when 2 is strong and Fight when Player 2 is weak. In this case, Player 1 will get a payoff of 0 when 2 is strong and 100 when Player 2 is weak. Expected payoff is then 0p+100(1-p)=100(1-p).

What is a spy worth? If p>1/2, knowing Player 2’s type increases profits from 0 to 100(1-p), so Player 1 would be willing to pay a spy up to 100(1-p). If p<1/2, knowing Player 2’s type would increase Player 1’s profits from 100-2p to 100-p, so Player 1 would be willing to pay a spy up to 100(1-p)- 100(1-2p)=100p.

Value of spy’s information as a function of p p 0 1 1/2 Value of information

Private Information and Auctions

Auction Situations Private Value – Everybody knows their own value for the object – Nobody knows other people’s values. Common Value – The object has some ``true value’’ that it would be worth to anybody – Nobody is quite sure what it is worth. Different bidders get independent hints.

First Price Sealed Bid Auction with private values Suppose that everyone knows their own value V for an object, but all you know is that each other bidder has a value that is equally likely to be any number between 1 and 100. A strategy is an instruction for what you will do with each possible value. Let’s look for a symmetric Nash equilibrium.

Case of two bidders. Let’s see if there is an equilibrium where both bidders bid some fraction a of their values. Let’s see what that fraction would be. Suppose that you believe that if the other guy’s value is X, he will bid aX. If you bid B, the probability that you will be the high bidder is the probability that B>aX. The probability that B>aX is the probability that X<B/a.

Two bidder case We have assumed that the probability distribution of the other guy’s value is uniform on the interval [0,100]. For number X between 0 and 100, the probability that his value is less than X is just X/100. The probability that X<B/a is therefore equal to B/(100 a). This is the probability that you win the object if you bid B.

So what’s the best bid? If you bid B, you win with probability B/(100a). Your profit is V-B if you win and 0 if you lose. So your expected profit if you bid B is (V-B) times B/(100a)=(1/100a)(VB-B 2 ). To maximize expected profit, set derivative with respect to B equal to zero. We have V-2B=0 or B=V/2. This means that if the other guy bids proportionally to his value, you will too, and your proportion will be a=1/2.

Bid-shading in this example In a first-price sealed bid private value auction, where each believes that the other’s value is uniformly distributed on the interval [0,100], there is a Bayes-Nash equilibrium in which each bidder bids half of her value.

Oil Lease Sales

Single bidder version A new oil field has come up for lease. The current owner has done no exploration of the oilfield. All that the current owner knows is that the field has two halves such that each half is worth either $0 or $3 million dollars with independent probabilities of ½ for each value for each half. It is common knowledge that there is only one possible buyer and that buyer has learned the value of one half of the field but not the other.

Bayes-Nash equilibrium Strategy for Seller--Posts a price. Strategy for Buyer—Rule that specifies whether she will buy or not at each possible price if the side she knows is worth $0 and if it is worth $3 million. Buyer tries to maximize her expected profit. Buyer will buy if and only if the price is less than or equal to her expected value, given her information. When seller posts his price, he doesn’t know whether the side the buyer explored is worth 0 or $3 million.

Extensive Form of simplified game Buyer sees 0Buyer sees 3 Nature Seller P=1.5-e P=4.5-e Buy Don’t Buy Don’t 1.5-e e e e e 3+e 4.5-ee4.5-ee Buyer Top number is Seller’s Profit. Bottom is Buyer’s Profit. e is a small number <1.

Trimming tree with subgame perfection Buyer sees 0Buyer sees 3 Nature Seller P=1.5-e P=4.5-e Buy Don’t 1.5-e e e 3+e 4.5-ee4.5-ee Buyer

What is the probability That the Buyer will buy the oil field if the price is 1.5-e? That the Buyer will buy the oil field if the price is 4.5-e?

Clicker question If the seller posts a price of just under $4.5 million, her expected profit is closest to. A)$1 million B)$1.5 million C)$2 million D)$2.25 million E)$3 million

Clicker question If the seller posts a price of just under $1.5 million, her expected profit is closest to A)$.5 million B)$.75 million C)$1 million D)$1.25 million E)$1.5 million

In a Bayes-Nash Equilibrium with a single buyer A) Seller will post a price of (just under) $4.5 million. Buyer will buy if buyer’s side is worth $3 million and will reject if her side is worth $0. B) Seller will post a price of (just under) $1.5 million. Buyer will buy whether her side is worth $3 million or 0. C) Seller will post a price of (just under) $1.5 million. Buyer will buy only if her side is worth $3 million.

In Bayes-Nash equilibrium for this game, the expected revenue of the seller is A)$1 million B)$2 million C)$2.25 million D)$3 million E)$4.5 million

What if there are two bidders? Each has explored a different half of the oil field and knows the value of the half she explored. The value of each side is either $3 million or 0, which depended on the flip of a fair coin. Total value of field is the sum of the two sides Each bidder knows what her side is worth, but not the other bidder’s side.

Posted Price The seller posts a price. If one buyer accepts and the other declines, oilfield is sold to that bidder at posted price. If both buyers accept, the seller tosses a fair coin to decide which buyer to sell to. If neither buyer accepts, oilfield remains undeveloped and seller and both buyers receive 0 payoff.

Would you buy? Suppose there are two buyers as described above and suppose that the seller posts a price of $4.1 million. If the half of the oil field that you examined is worth $3 million, would you buy it? A) Yes B) No

Expected values? Suppose there are two bidders and both would offer to buy at a posted price of $4.1 million if they saw $3 million and not if they saw 0 (Why not? We found that your expected value of the oilfield is $4.5 if you saw $3 million.) But wait... If the oilfield is worth 0, you get it for sure. But if it is worth $6 million, you get it with probability ½ because this means the other guy also bid $4.5.

A winner’s curse? So if you offer to buy at $4.1 million, then even though the oilfield is equally likely to be worth $6 million or $3 million, you are more likely to get it if its worth 3 than if its worth 6. We want to find the expected value of the oilfield conditional on your getting it if you offer to buy.

Conditional Probability P(A|B) =P(A and B)/P(B)) (Bayes’ rule) Let Event A be other guy saw $3 million. Let Event B be: Your offer is accepted. P(A and B) = ¼. P(B)=1/4+1/2=3/4. P(A|B)=1/4÷3/4=1/3. If you get the object, the probability that the other guy saw $3 million is only 1/3. Probability he saw 0 is 2/3.

Conditional expected value and winner’s curse Suppose that you saw $3 million on your side. Although the probability that the other guys saw $3 million is ½ on his side, the probability that he saw $3 million, given that you get the object, is only 1/3. The conditional expected value of the object given that you get it is therefore only $6 million x 1/3 + $3 million x 2/3= $4 million.

Winner’s curse in common value auctions If you win something in an auction or other bidding contest, makes it more likely that the object is of low value to other bidders. If low value to other bidders implies lower value to you, then it is a mistake to bid as much as your expected value given only your own information.

Possible equilibrium? Suppose the seller sets price at (just under) $4 million. Buyers will offer to buy if they saw $3 million. Buyers will not offer to buy if they saw $0. You will get a sale if at least one of the buyers saw $3 million.

Bayes-Nash equilibrium with two buyers There would be a Bayes-Nash equilibrium in which seller sets price at $4 million and buyers will offer to buy if their side is worth $3 million and will not offer to buy if their side is worth 0. Expected revenue of seller would then be (1/4)x0+(3/4)x4 million= $3 million

One more auction First-bidder, sealed bid auction. Two bidders. Object goes to high bidder. If there is a tie, coin flip decides who gets it. Each knows his own value, which is either 3, 5, or 8. Each believes that other’s value is 3 with probability.3, 5 with probability.3, and 8 with probability.4.

Checking an Equilibrium Candidate Is there a symmetric Bayes-Nash equlibrium in which each bidder bids 2 if his value is 3, 3 if his value is 4, and 4 if his value is 8? To check this out, suppose that the other bidder is using this strategy. See if this is then the best strategy for you. Recall that each believes that other’s value is 3 with probability.3, 5 with probability.3, and 8 with probability.4.

What is probability my bid wins? If I bid less than 2, I never win. If I bid 2, I win only if other bidder has value 3 and I win coin toss. So probability of winning if I bid 3 is always (1/2)x.3=.15 If I bid 3, I win if other bidder bids 2 or if other bidder bids 3 and I win coin toss. This happens with probability.3 +(1/2)x.3=.45. If I bid 4, I win if other bids 2 or 3 or if he bids 4 and I win coin toss. This happens with probability.3+.3+(1/2)x.4=.8 If I bid more than 4, I get it for sure.

Table of winnings My Bid Prob I get itExpected payoff: V=3 Expected payoff:V=5 Expected payoff: V= x.15=0.15*3x.15=0.456x.15= x.45=02x.45=0.9*5x.45= x.8=-0.81x.8=0.84x.8=3.20* 51-2x1=-20x1=03x1=3.0

Checking it out Look at table on previous slide. Note that if other guy is bidding 2 when his value is 3, 3 when his value is 4, and 4 when his value is 8, your best choices are: – 2 if 3 – 3 if 5 – 4 if 8 So there is a symmetric Bayes Nash equilibrium where each bidder uses this strategy.

Another Bayes Nash equilibrium? Suppose each bids 2 if 3, 3 if 5 and 5 if 8. What are your probabilities of winning? Same as before with bids of 1, 2, or 3, but not the same with bids of 4 or 5.

If other bids 2 if 3, 3 if 4, and 5 if 8 My BidProb I get it Expected payoff: V=3 Expected payoff:V=5 Expected payoff: V= x.1=0.15 *3x.15=0.456x.15= x.45=02x.45=0.90*5x.45= x..6= -.6 1x.6=.6 4x.6 =2.4* x1= -20x.45=03x.8=2.4* 61-3x1= -3-1x1=-12x1=2

Equilibrium So we see from the table that if the other bids 2 when value is 3, 3 when value is 5 and 5 when value is 8, that it is a best response for you to bid 2 when your value is 3, 3 when your value is 5 and 5 when your value is 8. Each is doing a best response to the other’s action. So this is also a symmetric Nash equilibrium.

Going, Going,…