Signaling Game Problems Problem 1 p 348 Quality

Signaling Game Problems

Problem 1, p 348 Quality Probability Value to Seller Value to Buyer Good Car q 10, 000 12, 000 Lemon 1 -q 6, 000 7, 000 If Buyers believe that the fraction of good cars on market is q, their Expected Value of a random car is 12000 q+7000(1 -q)=7, 000+5, 000 q • In this case, we can expect all used cars to sell for about PU=7, 000+5, 000 q. • If q>3/5, then PU=7000+5000 q> 10, 000 and so owners of lemons and of good cars and of will be willing to sell at price PU. • Thus the belief that the fraction q of all used cars are good Is confirmed. We have a pooling equilibrium.

There is also a separating equilibrium Quality Probability Value to Seller Value to Buyer Good Car q 10, 000 12, 000 Lemon 1 -q 6, 000 7, 000 Suppose that buyers all believe that the only used cars on the market. Then they all believe that a used car is only worth $7000. The price will not be higher than $7000. At this price, nobody would sell his good car, since good used cars are worth $10, 000 to their current owners. Buyer’s beliefs are confirmed by experience. This is a separating equilibrium. Good used car owners act differently from lemon owners.

Problem 3, page 348 • Suppose that buyers believe that product with no warranty is low quality and that with warranty is high quality. • High quality items work with probability H and low quality items work with probability L. Consumer values a working item at V. • Buyers are willing to pay up to LV an item that works with probability L. • Buyers are willing to pay up to V for any item with a money back guarantee. (If it works, their net gain is V-P and if it fails they get their money back so their net gain is 0. Therefore they will buy if P<V. )

Equilibrium • If the item with warranty sells for just under V and that with no warranty sells for just under LV, buyers will take either one. • Given these consumer beliefs, V is the highest price that sellers can get for high quality with warranty and LV is the highest price for the low quality without warranty. • Seller’s profits from high quality sales with guarantee are h. V-c and profits from low quality without guaranty are LV -c. • If seller put a guarantee on low quality items and sold them for V, his profit would be LV-c, which is no better than he does without a guarantee on these.

Equilibrium • If buyers believe that only the good items have guarantees, the Nash equilibrium outcome confirms this belief. • If fraction of items sold that are of high quality is r, then retailer’s average profit per unit sold Is r. HV+(1 -r)LV. • Retailer can not do better with a pooling equilbrium in which he guaranteed nothing, or in one in which he guaranteed everything. Can you show this?

Problem 5, page 350 George Bush and Saddam Hussein

The story • Bush believes that probability Hussein has WMDs is w<3/5. • When is there a perfect Bayes-Nash equilibrium with strategies? • Hussein: If WMD, Don’t allow, if no WMD allow with probability h. • Bush: If allow and WMD, Invade. If allow and no WMD, Don’t invade, If don’t allow, invade with probability b.

Payoffs for Hussein if he has no WMDs Payoff from not allow is 2 b+8(1 -b)=8 -6 b Payoff from allow is 4 b Hussein is indifferent if 4 b=8 -6 b or equivalently b=4/5. So he would be willing to use a mixed strategy if he thought that Bush would invade with probability 4/5 if Hussein doesn’t allow inspections.

Probability that Hussein has WMD’s if he uses mixed strategy • If Hussein does not allow inspections, what is probability that he has WMDs? • Apply Bayes’ law. P(WMD|no inspect)= P(WMD and no inspect)/P(no inspect)= w/(w+(1 -w)(1 -h))

Bush’s payoffs if Hussein refuses inspections • If Bush does not invade: 1 w/(w+(1 -w)(1 -h)) +9(1 -(w/(w+(1 -w)(1 -h))) • If Bush invades: 3 w/(w+(1 -w)(1 -h)) +6(1 -w/(w+(1 -w)(1 -h)) Bush will use a mixed strategy only if these two payoffs are equal. We need to solve the equation 1 w/(w+(1 -w)(1 -h)) +9(1 -(w/(w+(1 -w)(1 -h))) =3 w/(w+(1 -w)(1 -h)) +6(1 -w/(w+(1 -w)(1 -h)) for h.

Solution • Solving equation on previous slide, we see that if Saddam refuses inspections, Bush is indifferent between invading and not if h=35 w/3(1 -w). (Remember we assumed w<3/5) so 0<h<1) • If Saddam has no WMD’s, he is indifferent between allowing and not allowing inspections Bush would invade with probability 4/5 if there are no inspections.

Describing equilibrium strategies Saddam: Do not allow inspections if he has WMD. Allow inspections with probability h=3 -5 w/3(1 -w) if he has no WMD. (e. g. if w=1/2, h=1/3. If w=1/3, h=2/3. ) Bush: Invade if Saddam has WMD and allows inspections, Don’t invade if Saddam has no WMD and allows inspections. Invade with probability 4/5 if Saddam does not allow inspections.