Download
1 / 27
E N D
Lecture 9Strategic Uncertainty In fact not all games support a pure strategy Nash equilibrium. This leads players to randomize over their actions, or play mixed strategies. This randomization is called strategic uncertainty. It is collectively but non-cooperatively induced by the players rather than the information technology. In such situations we can predict the probabilities that players will move in certain ways but not be certain. The probabilities form the basis for a mixed strategy Nash equilibrium. If a game does not have a pure strategy Nash equilibrium it certainly has a mixed strategy Nash equilibrium.
Matching pennies Not every game has a pure strategy Nash equilibrium. In this zero sum game, each player chooses and then simultaneously reveals the face of a two sided coin to the other player. The row player wins if the faces on the coins are the same, while the column player wins if the faces are different.
The chain of best responses • If Player 1 plays H, Player 2 should play H, but if 2 plays H, 1 should play T. • Therefore (H,H) is not a Nash equilibrium. • Using a similar argument we can eliminate the other strategy profiles as being a Nash equilibrium. • What then is a solution of the game?
Avoiding losses in the matching pennies game • If 2 plays Heads with probability greater than 1/2, then the expected gain to 1 from playing Tails is positive. • Similarly 1 expects to gain from playing Heads if 2 plays Heads more than half the time. • But if 2 randomly picks Heads with probability 1/2 each round, then the expected profit to 1 is zero regardless of his strategy. • Therefore 2 expects to lose unless he independently mixes between heads and tails with probability one half.
Mixed strategy equilibrium the matching pennies game • Suppose 2 ( or the subjects assigned to play 2 on average) plays Heads half the time. Then a best response of 1 is to play Heads half the time. • Suppose 1 ( or the subjects assigned to play 1 on average) plays Heads half the time. Then a best response of 2 is to play Heads half the time. • Hence each player playing Heads half the time is a mixed strategy equilibrium, and since we have already checked there are no others, it is unique.
Taxation • Consider the problem of paying and auditing taxes. • The taxpayer owes 4, but might reduce his taxes by 2 through negligent accounting and 0 through fraud. • It costs the IRS 1 to check for irregularities (which uncover some irregularities) and 2 to uncover fraud. • The penalties are harsh, 2 for negligence, and 8 for fraud.
Monitoring by the collection agency • Equating the expected utility for the collection agency: 1011+ 4 12 + 2(1- 11- 12)= -11+ 512 + 3(1- 11- 12) and: 1011 + 4 12 + 2(1- 11- 12)= 212 + 4(1- 11- 12) • Solving these equations in two unknowns we obtain the mixed strategy: 11 =1/12=0.083 12 = 1/4 =0.250 13 = 2/3 =0.667
Cheating by the taxpayer • Equating the expected utility for the taxpayer across the different choices: -1221 = -621 - 622 - 2(1- 21- 22) and -1221 = -4 • Defining the only strategy that leaves the taxpayer indifferent between all three choices is therefore: 21 =1/3 =0.333 22 = 1/6 =0.167 23 = ½ = 0.500
Mixed strategy Nash equilibrium in the taxation game 21=0.333 22=0.167 23=0.50 11=0.083 12=0.25 13=0.667
Solution to the Ware case • If Ware enters with probability p = 0.734, then a best response of National is to enter with probability q = 0.633. • If National enters with probability q = 0.633, a best response of Ware is to enter with probability p = 0.734. • Therefore the strategy profile p = 0.734 and q = 0.633 is a mixed strategyNash equilibrium. • In this case the equilibrium is the unique.
Marketing groceries • In this simultaneous move game the corner store franchise would suffer greatly if it competed on the same feature as the supermarket. • This is illustrated by the fact that its smallest payoffs lie down the diagonal.
Strategies dominated by a mixture • The supermarket's hours strategy is dominated by a mixture of the price and service strategies. • Let π denote the probability that the supermarket chooses a price strategy, and (1-π) denote the probability that the supermarket chooses a service strategy. • This mixture dominates the hours strategy if the following three conditions are satisfied: π65+(1-π)50 > 45 or π > -1/3 π50+(1-π)55 > 52 or 3/5 > π π60+(1-π)50 > 55 or π > ½ • Hence all mixtures of π satisfying the inequalities: ½ < π < 3/5 dominate the hours strategy.
Existence of Nash equilibrium • Consider any finite non-cooperative game, that is a game in extensive form with a finite number of nodes. • If there is no pure strategy Nash equilibrium in the strategic form of the game, then there is a mixed strategy Nash equilibrium. • In other words, every finite game has at least one solution in pure or mixed strategies.
Strategic uncertainty • Strategic uncertainty arises when the solution, is a Nash equilibrium with a mixed strategy. • The uncertainty in equilibrium is directly attributable to the players’ choices rather than uncertainty about the environment. • For example in the taxation game all the players were playing a mixed strategy.
Quality control • Manufacturers do not consistently produce flawless products despite legions of consultants who have advised them against this policy. • Retailers help guard against flawed products by returning some of the defective items sent, and lending their brand to the ones they retail. • Consumers cannot judge product quality as well as retailers and producers, since each one experiences only a tiny fraction of the end product. • What is an acceptable defect rate, how often should retailers return defective items, and what are the implications for consumer demand?
TQM in strategic form • There are two strategies for each player. Having derived the strategic form of the game, we can easily locate the pure strategy Nash equilibriums. • There is a unique pure strategy Nash equilibrium, in which the producer only manufactures flawless products, the retailer only sells flawless products and the customer always buys the product.
Why is the pure strategyNash equilibrium unconvincing? • But is this Nash equilibrium convincing? • Sure you can’t eliminate any dominated strategies. • If, however, the producer does manufacture a defective item, the retailer, but not the consumer will know, and makes more by offering the item for sale. • Can the retailer convince consumers that they really will return defective products?
Is there a mixed strategy equilibrium? • Let q denote the probability that the retailer offers a defective product item sale. • Let r denote the probability the customer buys the item. • Let p be the probability of producing a flawless item.
Solving for r, the probability of buying • If 0 < q < 1, then the retailer is indifferent between offering a defective product and returning it. • In that case: 3r - 2(1 - r) = -1 ⇒ 3r – 2 + 2r = -1 ⇒ 5r = 1 ⇒ r = 0.2
How to solve for p and q • Once we substitute for r = 0.2 in the shopper’s decision, we are left with the diagram: • q is chosen so that the producer is indifferent between production methods; • p is chosen so that the shopper is indifferent between buying and not buying.
Solving q,the probability of offering the product • The producer will only mix between defective and flawless items if the benefit from both are equated: [6r + (1 - r)]q - 3(1 - q) = [3r + (1- r)] ⇒ 2q – 3 + 3q = 1.4 ⇒ 5q = 4.4 ⇒ q = 0.88
Solving for p, the probability of producing a flawless product • Investigating the cases above shows that in a mixed strategy equilibrium r = 0.2 and q = 0.88. • Since the shopper is indifferent between buying the item versus leaving it on the shelf, there are no expected benefits of acquiring the item: 9p - 10(1 - p)q = 0 ⇒ (9 +10q)p = 10q ⇒ p = 44/89
Offering a partial refund We now modify the game slightly. If the customer buys a defective product, she receives partial compensation.
A different outcome • In this case the manufacturer has a weakly dominant strategy of specializing in the production of flawless goods. • Recognizing this, the shopper picks a pure strategy of buying. • Realizing that the shopper will buy everything she is offered, the retailer never returns its merchandise to the manufacturer (and indeed there is never any reason too).
Summary • Not every game can be solved using the principle of iterated dominance. • Moreover not every game supports a pure strategy Nash equilibrium. • But every game does have at least one Nash equilibrium in pure or mixed strategies. • Strategic uncertainty arises when the solution to the game is a mixed strategy. In such games the players create the uncertainty their own optimizing decisions.
