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Normal Form Games of Incomplete Information SI 563 Lecture 6 Normal Form Games of Incomplete Information Professor Yan Chen Fall 2008 Some material in this lecture drawn Maggie Levenstein, Ross School of Business

Agenda Games of incomplete information Random events and incomplete information Risk and incentives in contracting Bayesian Nash equilibrium Lemons and Auctions

Games of Incomplete Information To say that a game is of incomplete information is to say something about what is known about the circumstances under which the game is played Games having moves of nature that generate asymmetric information between the players Type: different moves of nature that a single player privately observes

Examples Online auctions Russian roulette Oligopoly unrealistic to assume that bidders know the other bidders’ valuations or risk attitudes Russian roulette Both players have an incentive to pretend to be more reckless than they actually are Oligopoly Unrealistic to assume that one firm knows the cost structure of the other firm

Main Tools Harsanyi’s theory of incomplete information offers a means to get a handle on such matters: a technique for completing a structure in which information is incomplete Main technique: expected utility calculation

Solution Concepts: A Comparison Normal Form Games Extensive Form Games Nash Equilibrium Subgame Perfect Nash Equilibrium Bayesian Nash Equilibrium (Perfect Bayesian Equilibrium) Complete Information Incomplete Information

Random Events and Incomplete Information (Watson Chapter 24)

Example: The Gift Game 1,1 A 1F 2 NF GF 0,0 Friend (p) R -1,0 A 1,-1 Enemy (1 – p) 0,0 NE GE 1E R -1,0 Chance node: nature’s decision node; Nature determines player 1’s type: Friend (with probability p) or Enemy (1-p); Player 1 observes Nature’s move, so he knows his own type; Player 2 does not observe player 1’s type.

The Gift Game in Bayesian Normal Form 2 1 GF GE 1,2p-1 -1,0 p,p -p,0 1-p,p-1 p-1,0 0,0 Strategy: If F, G; if E, G GF NE NF GE NF NE In games of incomplete info, rational play require a player who knows his own type to think about what he would have done had he been another type.

Example: A Game of Incomplete Information 2 1 C D A x,9 3,6 6,0 6,9 12 with probability 2/3 x = B 0 with probability 1/3 Player 1’s payoff number x is private information; Player 2 knows only that x=12 with probability 2/3 and x=0 with probability 1/3. This matrix is not the true normal form of the game.

Extensive-form Representation 12,9 C 2 A12 D 3,6 1 C 6,0 x = 12 B12 (2/3) D 6,9 C 0,9 (1/3) A0 x = 0 1 D 3,6 C 6,0 B0 D 6,9

Normal-form Representation 2 1 C D 8,9 3,6 10,6 4,7 4,3 5,8 6,0 6,9 A12 A0 A12 B0 B12 A0 B12 B0 Player 1’s decision: (1) whether to select A or B after observing x=0; (2) whether to select A or B after observing x=12.

Risk and Incentives in Contracting Nature moves at the end of the game … - the simplest case (Watson Chapter 25)

Background Definitions More than one possible outcome can occur. Probability refers to the likelihood that an outcome will occur. Objective probabilities Subjective probabilities The expected value (average, mean) of a random variable is a weighted average of the values of all possible outcomes, with the probabilities of each outcome used as the weights . Variance and standard deviation are measures of dispersion of individual outcomes from the mean. Probabilities can either be objective or subjective. Objective: there is a 0.25 probability that from a generic deck of cards I will draw a club. Subjective: there is a 0.01 probability that I could beat the fifth-best basketball player in this room in a game of one-on-one. There is nothing inherently “inferior” or less precise about a subjective probability. (For example, an insider at a major corporation may have a subjective probability on the firm’s future stock return that is better than objective forecasts by outside analysts.) Subjective probabilities -- arise because of different information or differing means for processing it. A random variable is a variable that can take on more than one value or outcome. Examples: Flipping a coin: heads or tails. (discrete) Michigan football: win, lose, or tie. (discrete) 1994 IBM stock return: - 5%, +3%, +10%, ... (continuous) We will limit our attention to discrete random variables. 13

Expected Value and Variance Expectation: E(X) = S pi(Xi) = p1(X1) + p2(X2) + ... + pn(Xn), where pi = probability of outcome i, Xi = value of random variable associated with outcome i, and p1 + p2 + ... + pn = 1. Variance: n s2 = S pi[Xi - E(Xi)]2 i=1 Standard Deviation: s Expected value is a measure of central tendency and variance and standard deviation are measures of dispersion. Variance and standard deviation are often used as measures of risk. Example: Consider an offshore oil venture that has two possible outcomes: “success” and “failure.” If the venture succeeds, then profits equal $150 per share. If it fails, profits equal -$25 per share. There is a 0.15 probability of success and a 0.85 probability of failure. Hence, Expected value = (0.15)($150) + (0.85)(-$25) = $1.25 Variance = s2 = (150 - 1.25)2(0.15) + (- 25 - 1.25)2(0.85) = $3,905 15

Attitude Toward Risk Risk neutral individuals maximize expected value Everyone maximizes expected utility St Petersburg paradox: A gamble of consecutive tossing of a fair coin Payoff doubles for every consecutive heads that appears

Expected Utility Example The Bet: 50% chance of winning $100 50% chance of winning $400 Risk Neutral Player: EV = (.5)(100) + (.5)(400) = 250 Bet is equivalent to having $250 for sure Player would be willing to pay up to $250 for a lottery ticket with these odds Player would be willing to pay up to $250 for insurance rather than assume the risk of any bet that is worse than this bet

Expected Utility Example The Bet: 50% chance of winning $100 50% chance of winning $400 Risk Averse Player might have U = (I)½ EU = (.5)(100)½ + (.5)(400)½ = 15 What income will give him U = 15 for sure? 15 = (I) ½  I = 225 Bet is equivalent to having $225 for sure Player would be willing to pay up to $225 for a lottery ticket with these odds Player would be willing to pay up to $225 for insurance rather than assume the risk of any bet that is worse than this bet

Risk Aversion and Expected Utility If most of us are risk averse (in most situations), then we are obviously not trying to maximize the expected dollar return. Economist’s view: We maximize Expected Utility Utility Utility Curve Utility from $20,000 Utility from $10,000 Income $1,000s 10 20 30 40 50 Risk aversion implies a “concave” utility function, or “diminishing marginal utility of money.” Source: Maggie Levenstein, Ross School of Business 19

Risk Aversion and Expected Utility If most of us are risk averse (in most situations), then we are obviously not trying to maximize the expected dollar return. Economist’s view: We maximize Expected Utility Utility Utility Curve Expected utility from a bet with 50-50 odds of 10 and 20 Income $1,000s 10 20 30 40 50 Source: Maggie Levenstein, Ross School of Business 19

Utility (15) > .5 U(10) + .5 U (20) Risk Aversion and Expected Utility If most of us are risk averse (in most situations), then we are obviously not trying to maximize the expected dollar return. Economist’s view: We maximize Expected Utility Utility Curve Expected utility of 15 for sure Utility (15) > .5 U(10) + .5 U (20) Expected utility from a bet with 50-50 odds of 10 and 20 Income $1,000s 10 20 30 40 50 Source: Maggie Levenstein, Ross School of Business 19

Example: Lottery or Sure Thing? Receive $950 A You H Receive $2000 (1/2) B (1/2) Receive $0 T A: sure thing; B: lottery

Utility functions and concavity v(x) v(1000) Concave utility function v(0)/2+v(2000)/2 x v(0) 1000 2000 v(x): utility of receiving x dollars

Linear Utility Function: Risk Neutral v v(2000) v(x) Linear utility function v(1000) = v(0)/2+v(2000)/2 x v(0) 1000 2000

Different Levels of Risk Aversion v(x) Risk neutral Slight risk aversion Greater risk aversion x

A Principal-Agent Game with Moral Hazard Set of players: {Pat, Allen} Pat: Principal, risk neutral, v(x) = x Allen: Agent, risk averse, v(x) = x^a Pat: write a contract, (wage, bonus) Allen: exert high or low effort Success depends on Allen’s high effort as well as a random factor

Extensive Form (w -1)a (w + b -1)a U H A w,b P A 2 – w, (1/2) (1/2) 2 – w, wa w,b Y P L A N 0, 1 w: wage b: bonus, paid only if successful High effort cost: 1

Principle-Agent Game with Expected Payoffs 2 –w + (4 - b)/2, H w,b (w + b -1)a /2 + (w - 1)a/2 A P Y 2 – w, wa A L N 0, 1 Pat would like Allen to exert high effort. Can she write a contract that induces it?

Case 1: No-bonus Contract If Pat sets b=0 Allen has no incentive to exert high effort The best Pat can do is to offer w = 1 Allen is willing to accept Best no-bonus contract: w=1, b=0 Payoff vector (1, 1) P&R: “Used cars sell for much less than new cars because there is asymmetric information about their quality. The seller of the used car knows much more about the car than the prospective buyer does.”

Case 2: Bonus Contract (solution via backward induction) Last step: Incentive-compatibility constraint Allen’s expected payoff from H has to be at least as great as his payoff from L Second to last step: Individual rationality (or voluntary participation) constraint Allen’s expected payoff from H has to be at least as great as his outside option The best contract has to satisfy the IC and IR constraints with equality Solution: w = 1, b = 2^(1/a) P&R: “Used cars sell for much less than new cars because there is asymmetric information about their quality. The seller of the used car knows much more about the car than the prospective buyer does.”

Bayesian Nash Equilibrium and Bayesian Rationalizability (Watson Chapter 26)

Finding BNE Method 1 Method 2 Write down Bayesian normal form Solve for Nash equilibrium of the normal form: Bayesian Nash equilibrium Or, solve for the set of strategies which survive iterated elimination of dominated strategies: Bayesian rationalizability Method 2 Treat types as separate players (omit)

Example: A Game of Incomplete Information 2 1 C D A x,9 3,6 6,0 6,9 12 with probability 2/3 x = B 0 with probability 1/3 Player 1’s payoff number x is private information; Player 2 knows only that x=12 with probability 2/3 and x=0 with probability 1/3. This matrix is not the true normal form of the game.

Extensive-form Representation 12,9 C 2 A12 D 3,6 1 C 6,0 x = 12 B12 (2/3) D 6,9 C 0,9 (1/3) A0 x = 0 1 D 3,6 C 6,0 B0 D 6,9

Normal-form Representation 2 1 C D 8,9 3,6 10,6 4,7 4,3 5,8 6,0 6,9 A12 A0 A12 B0 {B12B0, D} is the Bayesian rationalizable set, and the unique BNE. B12 A0 B12 B0 Iterated elimination of dominated strategies: (1) B12B0 dominates B12A0; A12B0 dominates A12A0. (2) D dominates C. (3) B12B0 dominates A12B0.

Lemons and Auctions (Watson Chapter 27)

Adverse Selection This is a problem of hidden characteristics (when one side of a transaction knows something about itself that the other does not) and self-selection. The uninformed party gets exactly the wrong people trading with it, so we say that the uninformed party gets an adverse selection of the informed parties. P&R: “Used cars sell for much less than new cars because there is asymmetric information about their quality. The seller of the used car knows much more about the car than the prospective buyer does.”

Incomplete Information and Adverse Selection worst risks are the ones most likely to buy insurance, pushing up the price for the best risks low-quality products can crowd out high-quality products There is a “market failure” because sellers of low-quality “lemons” impose a negative externality on the sellers of high-quality products. When low-quality products are offered for sale, they adversely affect the perceived value of high-quality products if buyers cannot differentiate low- and high-quality. Low-quality products prevent the market for high-quality products from functioning properly. These markets are interesting because there may be indirect ways for the uniformed party to infer what is going on Adverse selection also arises in insurance markets because the “worst risks” are most likely to buy. This “forces out” the good risks -- those who do not want to pay the high premiums necessary to cover the costs associated with pooling in the high risk buyers. But this, in turn, causes the insurer’s average costs to rise, forcing the average premium up, and driving away still more low or moderate risk customers. In the end, it is possible (under some circumstances) that only the highest-risk individuals receive insurance. Adverse selection arises whenever there is a hidden characteristics problem and people on the informed side of the market self-select in a way that is harmful to the uninformed side of the market.

Adverse Selection Adverse selection is also known as the lemons problem suppose you purchase a new car, drive it for 1 month, and then for reasons entirely out of your control, you must sell it what price do you think you could get for your car? Why? how could you minimize the problem? P&R: “Used cars sell for much less than new cars because there is asymmetric information about their quality. The seller of the used car knows much more about the car than the prospective buyer does.”

Lemon: An Example Jerry is in the market for a used car Freddie offers an attractive 15-year old sedan for sale Blue book value for the car is p The car is a peach with probability q If peach: worth $3000 to Jerry, $2000 to Freddie If lemon: worth $1000 to Jerry, 0 to Freddie What is the efficient outcome?

TP NP TL NL 0,2000 0, 2000 0,0 F F J J 3000-p, p 1000-p, p T T N N FIGURE 27.1 The lemons problem. F F TP NP TL NL J J 3000-p, p 0,2000 0, 2000 1000-p, p 0,0 T T N N Peach Lemon (q) (1 - q)

Extensive form Nature moves first Jerry and Freddie then choose their strategies simultaneously

Bayesian Normal Form How many strategies does Jerry have? How many strategies does Freddie have? What is the size of the matrix

2000q+1000-p, p (3000-p)q, pq 0, 2000q (1000-p)(1-q), 2000q+(1-q)p Freddie TPNL NPTL TPTL NPNL 2000q+1000-p, p (3000-p)q, pq (1000-p)(1-q), 2000q+(1-q)p 0, 2000q T N

BNE 1: Only Lemons Traded (T, NPTL): two conditions should hold (1) (1000-p)(1-q) ≥ 0, or 1000 ≥ p (2) 2000q+(1-q)p ≥ max{p, pq, 2000q} (non-binding) Intuition: If price is below $1000, F would only want to bring lemons to the market Anticipating that only a lemon will be for sale, Jerry is willing to pay no more than $1000

BNE 2: Both Lemon and Peach Traded (T, TPTL): two conditions should hold (1) 2000q+1000-p ≥ 0, or 2000q+1000 ≥ p (2) p ≥ max{pq, 2000q+(1-q)p, 2000q} or p ≥ 2000 (3) Combining both conditions: q ≥ ½ Intuition: Jerry’s expected value from owning the car exceeds its price; Freddie is willing to bring a peach to the market; The probability of a peach should be sufficiently high

Solving the Adverse Selection Problem Some limited ways to address this have a mechanic check over the car offer a warranty government “lemon laws” e.g. Wall Street Journal, 10/18/96: “California is prohibiting Chrysler Corp. from shipping vehicles into the state for 45 days as punishment for selling defective used vehicles, which allegedly should have been labeled as ‘lemons.’. . . In California, a new vehicle is considered a lemon once the owner has attempted to fix a defect four or more times. It is also deemed a lemon after it has spent 30 days or more in the repair shop over a 12-month period. Once it has been acknowledged as a lemon, auto makers are required to buy it back for the original purchase price. . .” establish a reputation Cure: signal high quality with warranties, certified mechanics, state lemon laws. SOLUTION #1: Provide medical policies to entire groups, e.g., through employers. Make coverage mandatory (diversifies population signing up for ins.) refuse coverage for “pre-existing conditions”, and limit choice. SOLUTION #2: Limit the amount of ins. available at most favorable odds. SOLUTION #3: Refuse to insure those people who are worst risks. SOLUTION #4: Each risk could be evaluated and appropriate price charged. Insurance co. now subject to claims that it is discriminatory. In insurance markets (e.g., life and auto), screen applicants carefully, and increase scrutiny as the size of the policy increases. Limit the amount of insurance offered at favorable rates. Avoid markets with many high-cost/low-quality participants. (Twentieth Century Insurance in California only offers auto policies to drivers who can verify that they are extremely low risk -- and until the 1994 earthquake the company did very, very well.) In products markets, use warranties and other measures to assure product quality. (Though one problem with this approach is that extended warranties would appeal most to buyers who expect to put the product through the most wear and tear -- in other words, the flip side of adverse selection among sellers is moral hazard among buyers). Finally, repeated interactions and brand name capital.

Some Cures for Adverse Selection in Providing Health Care Provide medical policies to entire groups (e.g., through employers), Make coverage mandatory Refuse coverage for “pre-existing conditions” Limit choice

Some Cures for Adverse Selection in Providing Health Care Example: Suppose a company offers 3 insurance options to employees an HMO at no cost a mid-range plan that has more physician choice & better coverage, but costs each employee $50 per month with higher deductibles a “Cadillac plan” that gives complete choice, wonderful benefits, & no deductibles but costs $150 per month How might different kinds of employees choose among plans?

Auctions: Background What’s so interesting about auctions? An alternative to bargaining for selling a fixed supply of a commodity for which there is no well-established, ongoing market. Applications Real estate, art, flowers, oil leases Privatization and deregulation Government contracts Electricity Airwaves: FCC spectrum Auctions Allocation of common resources E-commerce: eBay

Auctions: Background - cont. Auction Institutions English Dutch First Price sealed-bid Vickrey Google Adwords (position) Many other kinds

Types of Auctions Private value auctions Common value auctions Bidders’ valuations for the auctioned item(s) are independent from one another and are their private information. e.g., flowers, art, antiques. Common value auctions Bidders are uncertain about the ultimate value of the item, which is the same for all bidders. e.g., oil leases, Olympic broadcast rights. Affiliated (correlated) value auctions Bidders’ valuations for the auctioned item(s) are correlated, but not necessarily the same for all. In between private and common value auctions.

Auction Research Research Questions What do we know? Efficiency comparison of auction institutions Revenue comparison Bidder earning comparison Collusion? Transparency? What do we know? Single item: well Multiple items: little Substitutes Complements

Auction Research – cont. Agenda for theoretical research Multi-item auctions Agenda for experimental research Test/discriminate among theories Design and test new institutions

English Auction Background Commodities Oral auctions in English-speaking countries. Originally “Roman.” Commodities Antiques, artworks, cattle, horses, real estate, wholesale fruits and vegetables, old books, etc.

Rules for Experiment Auctioneer first solicits an opening bid from the group. Anyone who wants to bid should call out a new price at least $1 higher than the previous high bid. The bidding continues until all bidders but one have dropped out. The highest bidder gets the object being sold for a price equal to the final bid. Winner’s profit = Buyer Value – price; Everyone else’s profit = 0. Your Buyer Value = Last two digits of your SSN

English Auction Outcome Optimal strategy Participate until price = buyer value, then drop out. Equilibrium Outcome The highest bidder gets the object at a price close to the second highest Buyer Value. Comparative statics As n increases, the winning bid is closer to the highest BV. The more spread-out the different bidders’ valuations are, the larger |vmax-v2nd|. This means that if there is wide disagreement about the item’s value, the winner might be able to get it cheaply. Problems Collusion; bidding rings.

Dutch Auction Background Commodities Rules Wholesale produce, cut-flower markets in the Netherlands. Commodities Flowers in the Netherlands Fish market in England and Israel Tobacco market in Canada Rules Auctioneer starts with a high price. Auctioneer lowers the price gradually until some buyer shouts “Mine!” The first buyer to shout “Mine!” gets the object at the price the auctioneer just called Winner’s profit = Buyer Value – price; Everyone else’s profit = 0. Your Buyer Value = 100 – Last two digits of your SSN

First-Price Sealed-bid Auctions Background Used to award construction contracts (lowest bidder), real estate, art treasures; Rules Bidders write their bids for the object and their names on slips of paper and deliver them to the auctioneer. The auctioneer opens the bid and find the highest bidder. The highest bidder gets the object being sold for a price equal to her own bid. Winner’s profit = Buyer Value – price; Everyone else’s profit = 0. Your Buyer Value = First and second number of the last four digits of your SSN.

First-Price Sealed-bid Auctions – cont. Set up the problem: In a sealed-bid, first price auction in a private values environment with n bidders, each bidder has a private valuation, vi, which is his private information. The distribution of vi is common knowledge. Let Bi denote the bid of player i. Let π i denote the profit of player i. If vi ~ u[0,100], what is the Bayesian Nash equilibrium bidding strategy for the players? Optimal bidding strategies: If Bi≥vi, then π i ≤ 0. Therefore, Bi < vi, which gives: π i = 0 , if Bi ≠ maxj{Bj}, or π i = vi - Bj , if Bi = maxj{Bj} The question is how much below vi should his bid be? The less Bi is, the less likely he will win the object, but the more profit he makes if he wins the object.

First-Price Sealed-bid Auctions – cont. You are characterized by the strategy-type two tuple, (B,v). Suppose the other bidder’s value is X, and she is characterized by (αX,X), where α є (0,1). Your expected profit is: Eπ = P(Your bid is higher)·(v-B)+P(Your bid is lower)*0 With uniform distribution, P(X<B/α) = (1/100)(B/α). Therefore, Eπ=(1/100)(B/α)(v - B). assuming risk neutrality, you choose B to: max B(v – B) = Bv – B2 B It follows that B = v/2.

First-Price Sealed-bid Auctions – cont. With n bidders P(Your bid is highest) = [(1/100)(B/α)]n-1. max Bn-1(v – B) => B = [(n-1)/n]v. B Note: as n increases, B  v. i.e., increased competition drives bids close to the valuations. Equivalence of Dutch and First-price, sealed-bid auctions: same reduced form. The object goes to the highest bidder at the highest price. A bidder must choose a bid without knowing the bids of any other bidders. Optimal bidding strategies are the same.

Sealed-bid, Second-price Auctions Background: Vickrey (1961). Commodities stamp collectors’ auctions US Treasury’s long-term bonds Airwaves auction in New Zealand eBay and Amazon Rules Bidders write their bids for the object and their names on slips of paper and deliver them to the auctioneer. The auctioneer opens the bid and finds the highest bidder. The highest bidder gets the object being sold for a price equal to the second highest bid. Winner’s profit = Buyer Value – price; Everyone else’s profit = 0. Your Buyer Value = 100 – First and second number of the last four digits of your SSN.

Sealed-bid, Second-price Auctions – cont. Equilibrium bidding strategy: It is a weakly dominant strategy to bid your true value. Let V be your Buyer Value, let B be your bid, and let X be the highest bid made by anybody else in the auction. We want to show that overbidding or underbidding cannot increase your profit and might decrease it. Let πt be your profit when B = V. Let π be your profit otherwise.

Sealed-bid, Second-price Auctions – cont. Proof: First consider the case of overbidding, B>V. 1. X>B>V: You don’t get the object either way: π = πt = 0. 2. B>V>X: π = V – X = πt > 0. 3. B>X>V: π = V – X < 0, but πt = 0. Next consider the case of underbidding, B<V. 1. X<B<V: π = V – X = πt > 0. 2. B<X<V: π = 0, but πt = V – X > 0. 3. B<V<X: You don’t get the object either way: Equivalence of English and sealed-bid, 2nd Price. The object goes to the highest bidder. Price is close to the second highest BV.

Google Adwords Auction Generalized second-price auction (GSP) Sort bids Top x bids wins Bidder who wins the nth position pays the (n+1)th bids Is it VCG?

GSP: Properties Google’s “unique auction model uses Nobel Prize-winning economic theory to eliminate … that feeling that you’ve paid too much.”† GSP is not VCG when x > 1 Example † "Maximize Your Revenue From Search Results With Google AdSense”, Google 2004. (http://www.google.com/adsense/afs.pdf)

Homework Assignment (For Practice Only) Chapter 24: # 3 Chapter 25: #3, 5 Chapter 26: #1, 7 Chapter 27: #3, 4 (a, b)