A game is a collection of players, the actions those players can take, and their preferences over the selection of actions taken by all the players

Transcription

1 Game theory review A game is a collection of players, the actions those players can take, and their preferences over the selection of actions taken by all the players A strategy s i is dominant for player i if, given any selection of strategies a i by i s opponents, s i maximizes i s payoff If all the players are playing dominant strategies, they are playing a purestrategy Nash equilibrium of the game We d like to design games for which the players dominant strategies are simple, and the outcomes are desirable. Today we ll look at the quintessential example of such a game. What are auctions? They are games. Players: the bidders, denoted i = 1, 2,..., N Actions: their bids, b 1, b 2,..., b N Payoffs: if buyer i wins, he gets a payoff v i t i, where v i is bidder i s value and t i (b 1, b 2,..., b N ) is a payment (not necessarily his bid), and if buyer i loses, he gets a payoff of zero The English Auction This is the most important game in all of economics. The price clock starts at zero The auctioneer raises the price clock slowly, allowing agents to indicate whether they want to continue or withdraw When the second-to-last buyer withdraws from the auction, the winner s payment is set equal to the current price on the clock (call it b (2) ), and the winner is the buyer who failed to withdraw The English Auction How should bidders behave in the English auction? The English Auction Theorem 1. It is a dominant strategy to drop out of the English auction at b i = v i. If a buyer bids b i = v i, we say he bids honestly. 1

2 The English Auction The key to the auction is that you control the likelihood you win, but not the price you pay, which will always be the lowest price you could pay and still be a winner (meditate on this). What if i was to stay in until b i > v i? There s two cases: (1) i would have won when bidding b i = v i, and (2) i would have lost when bidding b i = v i. Case 1: if i would have won when bidding b i = v i, then the nexthighest bid is less than v i, so i makes the same payment when bidding b i > v i and i s payoff doesn t change. Case 2: if i would have lost when bidding b i = v i, then the nexthighest bid is greater than v i, so i makes a loss now that i wins with a bid above v i. In either case, i is weakly better off bidding honestly. The English Auction What if i was to bid b i < v i? There s two cases: (1) i would have won when bidding b i = v i, and (2) i would have lost when bidding b i = v i. Case 1: if i would have won when bidding b i = v i, then the nexthighest bid is less than b i, so i makes the same payment when bidding b i = v i and i s payoff doesn t change. Case 2: if i would have lost when bidding b i < v i, then the nexthighest bid is greater than b i but less than v i, so i could have gotten a strictly positive payoff instead of zero. In either case, i is weakly better off bidding honestly. If b i > v i and b i < v i are both dominated by bidding b i = v i, then b i = v i is a dominant strategy. The English Auction Why is the English auction so important? Dominant strategies: the buyers have a dominant strategy to bid honestly Efficiency: the buyer with the highest value wins Stability: if the price paid by the winner were any lower, some other buyer could rightfully object Individual rationality: since b i b (2), any winner s payoff is v i b (2) v i b i 0, so no buyer will regret winning Privacy preserving: if the auction ends then the buyer with the secondhighest value drops out, no one ever learns the winner s true value 2

3 Robust: players optimal strategies do not depend on their beliefs about their opponents It is easy to participate, and the outcome satisfies many desirable properties. The Second-price Auction The English auction is an open format: the buyers indicate their interest in participating over time Sometimes, this is infeasible or undesirable, and a closed or silent format is adopted It retains most of the positive features of the English auction, except, potentially, for privacy preservation of the winner s value The Second-price Auction In the second price auction (SPA), Each buyer i submits a bid b i The highest bidder wins, but pays the second-highest bid A buyer bids honestly if b i = v i. Theorem 2. It is a dominant strategy to bid honestly in the second-price auction. The Second-price Auction Charity auctions, for example, often use silent auctions to sell donations for cash, which are essentially second-price auctions Many Internet-based sales platforms use second-pricing, or a generalization of it (Google AdWords) Running an English auction has high transactions costs (Coase) What is a second-price auction? Let p i (b i, b i ) be the probability that i wins, given that he bid b i against opponents bidding b i. In the SPA, i s payment is constructed as t i (b i, b i ) = + p j (b i, b i )b j j i }{{} Welfare of the other agents if i participates; zero if i wins p j (b i =, b i )b j j i }{{} Welfare of the other agents if i were to opt out; second-highest bid/value if i wins 3

4 If i wins, the welfare of the other agents if i participates is zero, while the welfare of the other agents if i were to opt out is the second highest value when i wins; if i loses, i has no impact on welfare in either case, and makes a payment of zero. This is a generalization of Lindahl pricing; we call it Vickrey pricing, for the economist who figured out how to incorporate private information into the problem of selling goods. Other concerns Revenue maximization is this the best way to sell a good if I want to maximize profits or minimize costs? Multiple units what if the seller has multiple units and the buyers still only want one? What if the seller has multiple units and the buyers all would like multiple units? What if the seller has multiple heterogeneous goods and the buyers have different valuations over each? Let s look at revenue maximization first. Revenue maximization What s the worst thing that can happen to a seller in the SPA? Some buyer i submits a yuge bid, but......no one else does, so the good is sold for a very low price despite the buyer having a really high value for it. Basicaly, the seller inadvertently faces a monopsonist, and would be committed to trading at a price of zero. To protect sellers from bad outcomes like this, we add reserve prices. Revenue maximization In the second price auction with a reserve price (SPAR), The seller sets a reserve price r Each buyer i submits a bid b i The highest bid greater than the reserve price wins. The winner pays the maximum of the second-highest bid and the reserve price. A buyer bids honestly if b i = v i. Theorem 3. It is a dominant strategy to bid honestly in the SPAR. But is this the best way to sell a good? Can it raise the most revenue? 4

5 The First-price Auction To appreciate the profit maximization question, instead of the second-price auction, consider the first-price auction with reserve (FPAR): The seller sets a reserve price r Each buyer i submits a bid b i The highest bid greater than the reserve price wins. The winner pays his bid. Does this raise more or less revenue than the second-price auction? The First-price Auction In the FPA, buyers solve max b i p(b i )(v i b i ) where p(b i ) is the probability of winning, given a bid of b i Is honest bidding a good strategy? The FONC is p(b i )(v i b i ) }{{} Benefit of a higher likelihood of winning and getting a payoff p(b i ) b i }{{} Cost of paying a higher bid, conditional on winning = 0, just like a monopolist. (We ll skip over how to solve this exactly for the moment, trust me that I know how to do it) In the SPA, agents bid honestly (+) but only pay the second-highest bid (-). In the FPA, agents shade their bids down (-) but pay the highest bid (+). Which of these effects dominates? Simulations What happens to revenue as the number of bidders increases, in particular? 5

6 Revenue equivalence Theorem 4. Any method of selling a single good to buyers with independent, private values that gives the good to the agent with the highest value above the reserve price and charges the losers nothing raises the same amount of revenue. So the FPAR and SPAR with the optimal reserve price are both optimal in the class of all possible ways of selling things. Procurement auctions As a tool of public policy, auctions are incredibly popular. Suppose the government is trying to procure some good (like a bridge or computer or fighter plane) that it values at v. It wants to buy at the lowest price it can. The reverse auction or procurement auction is the game where i = 1, 2,..., N sellers each submit a bid b i The lowest bidder wins, and is paid the second-highest bid We can also impose a reserve price: if all the losing bids are above r, the winner is paid only r. The efficient reserve price would be that any winning bid must be less than v, so the government doesn t pay more than the good is worth. Collusion in auctions This assumes that the players in the game are not colluding or otherwise working together How should firms collude to maximize their payoffs in a standard SPAR? In the reverse auction? Porter and Zona (1992) considered collusion at procurement auctions in Long Island for highway construction projects, and look for this strategic over-bidding We all sat at the conference table one of the contractors would have a list of upcoming contracts... they d talk about the contract... how much money who won the last one... who should get this one... The contractors who were tagged to be the low bidders would work out their We all sat at the conference table one of the contractors would have a list of upcoming contracts... they d talk about the contract... how much money who won the last one... who should get this one... The contractors who were tagged to be the low bidders would work out their bid figures... The rest of the contractors would then come up with higher bids. 6

7 Collusion in auctions Porter and Zona look at the backlog of work the firm faces, and how intensely it is currenlty utilizing its capactiy: these firms should have higher marginal costs of an additional job, and should bid higher They find that firms bidding lower have a stronger correlation between cost shifters and their bid, while the bids made by firms who big higher have little connection with anything Unfortunately, if an antitrust authority or procurement agency were to publicly announce the adoption our test procedure, it would be relatively easy for an effective cartel to tailor its phantom bids to disguise collusive behavior. For example, all cartel firms could scale their competitive bids up by the same percentage. The bid ranking would then coincide with cost rankings. Other issues Risk aversion: favors the FPAR Learning from others bids: favors the SPAR Budget constraints on the bidders: favors the SPAR Regret aversion: favors the SPAR Summary from last time Auctions are important market designs: buyers or sellers submit bids, highest or lowest bids win, and winners make a payment that potentially depends on their bid or those of others The second-price auction with reserve (SPAR) is the game where (1) the seller sets a reserve price, (2) each buyer submits a bid, (3) the highest bidder above the reserve wins, but pays the maximum of the reserve price and the second-highest bid. The winner gets a payoff of v i max{r, b (2) }, and the losers get payoffs of zero It is a weakly dominant strategy to bid honestly in the SPAR, the outcome is efficient if r = 0, and profit-maximizing if the reserve price is set optimally The SPAR can easily be turned into an auction to buy rather than to sell: the lowest bidder wins, and receives the minimum of the second-lowest bid or the reserve price. This is great for governments who want to procure goods like construction projects. 7

8 Auction-like markets ebay looks like a second price auction: The seller can set a secret reserve price, buyers can make open bids or use a proxy bid (a robot that bids for them up to a certain limit), and the auction ends at a pre-specified time Bid-sniping: enter bids right at the end of the auction in order to steal the good from the current standing high bidder 40 percent of all ebay-computers auctions and 59 percent of all ebay- Antiques auctions as compared to about 3 percent of both Amazon-Computers and Amazon-Antiques auctions, respectively, have last bids in the last 5 minutes. The pattern repeats in the last minute and even in the last ten seconds. In the 240 ebay-auctions, 89 have bids in the last minute and 29 in the last ten seconds. In Amazon, on the other hand, only one bid arrived in the last minute. (Roth and Ockenfels, 2002) Auction-like markets In a penny auction or all-pay auction, we start at a price of zero. Each bidder pays a bid increment to stay in. Once all a bidder s competitors have dropped out, he is declared the winner and given the good This is a popular ecommerce business model It is totally evil. It is not an auction, it is a war of attrition. Auction-like markets The Better Business Bureau warns consumers, although not all penny auction sites are scams, some are being investigated as online gambling. BBB recommends you... know exactly how the bidding works, set a limit for yourself, and be prepared to walk away before you go over that limit. The idea is that the goods might sell for low prices: $30 for a tv, say. But hundreds or thousands of people each bid a few bucks on it, so the company is making a ton of money in the background. These losses add up for the poor people involved. It s really a lottery. Multiple unit auctions There are a lot of important multiple unit auctions, where the market designer is looking to sell or buy more than one unit at a time: h omehttp : // But in the SPAR, only one unit is sold or purchased at a time. We want to sell or buy more than one unit at a time. 8

9 Clean Power Plan Here s a case study that s in court right now: Using the Clean Air Act, the Federal government has set emissions targets for power generation for each state in the country that need to be met by 2022, 2029, and 2030 The reality of the law is that it requires each state to close a certain number of coal-burning power plants The Federal Government has advocated using cap-and-trade or other marketbased mechanisms Many actors have sued the Federal Government over this; regardless of how those lawsuits are decided, many actors will then sue the states when they try to implement their plans Clean Power Plan The expected discounted profit of a plant to owner i is π i, i = 1, 2,..., N; suppose they are all equally dirty, to keep the problem simple, but they have varying levels of investment that make some more profitable/efficient than others The market is individually rational only if the owner receives at least π i for agreeing to close his plant down; otherwise there s a lawsuit The government has told us we need to close down at least K plants, K < N We want to find the cheapest way to induce the least profitable plants to exit the market, such that we satisfy the Federal Government s emissions constraint Multiple units The second price auction gives us an important principle: if you control the likelihood you win, but not the price you pay, it s possible to give you a dominant strategy to bid honestly We can only implement the efficient outcome (closing down the least profitable plants) if we know the agents true valuations, which are only known to the agents themselves But we can extend the basic idea of the SPAR pretty easily here: all of the winners pay (receive) the bid of one of the losers 9

10 Highest-Rejected Bid (HRB) Auctions Suppose we want to sell K units to N buyers, N > K: All the buyers i = 1, 2,..., N submit a bid, b i. Order the bids from highest to lowest, b (1) b (2)... b (N). The, at most, K buyers with bids above a reserve price r win, but they all pay the maximum of the highest losing bid or the reserve price. A winner then gets a payoff of v i max{b (K+1), r}, where v i is i s value of winning the good. Notice, it is a dominant strategy to bid honestly in the HRB auction, just like the SPAR. Lowest-Rejected Bid (LRB) Auctions Suppose we want to buy K units from N sellers, N > K: All the sellers i = 1, 2,..., N submit a bid, b i. Order the bids from lowest to highest, b (1) b (2)... b (N). The, at most, K sellers with bids below a reserve price r win, but they all receive the minimum of the lowest losing bid or the reserve price. A winner then gets a payoff of min{b (K+1), r} c i, where c i is i s cost of providing the good. Again, it is a dominant strategy to bid honestly in the LRB auction. The Clean Power Plan How do we adapt the LRB to the CPP problem? So we can induce the least profitable/most inefficient plants to leave the market, and we implement the efficient outcome in dominant strategies. What is the new political problem that emerges? Thoughts? Strategies (how could I drive down power plants values of their firms without throwing the money away)? 10