Strategic Play for Markets and Auctions 45-975

1. Strategic Play for Markets and Auctions45-975 Professor Robert A. Miller Mini 4, 2005

2. Course Materials Course Website: http://www.comlabgames.com/45-975 Text: Robert A. Miller and Vesna Prasnikar Strategic Play, Stanford University Press, 2004 A draft can be found at: http://wpweb2k.gsia.cmu.edu/strategicplay/ user id: strategicplay password: draft The teaching assistant isJeremy Bertomeu with email at: bertomeu@cmu.edu

3. The first half of this lecture introduces the free form module for analyzing trading mechanisms, explains how limit order markets work, and provides some examples to show how prolific they are. These include retail markets, stock markets, and markets for derivative assets. • Most auctions are simple examples of limit order markets. The second half of this lecture describe the 5 main kinds of auctions. We conclude the lecture by analyzing how traders should bid in auctions, and how much revenue auctions generate when experienced bidders play their (equilibrium) strategies. Lecture 1ALimit Order Markets

4. Limit order markets • Anyone seeking to trade in the market must submit a market order or a limit order. • Each order is for a given quantity, negative (blue) quantities standing for units for sale, positive (red) for units demanded. • Limit orders also specify a transaction price. • All market transactions match a market order with one or more limit orders, and take place at the limit order price(s).

5. Precedence • Market orders to buy are matched against the lowest price limit order(s) to sell. • If two limit orders to buy are submitted at the same price, the order submitted first is matched against a market sell order before the more recently submitted buy order. • Similarly lower priced limit orders to sell have a higher priority than higher priced limit sell orders, and if two bidders seeking to sell a unit at the same price the person who bid first will be matched before his rival seller.

6. Trading window The trader in this market has just placed a sell order for 9 units at price 5,800, with an expiry time of 60,000 seconds (that is 16 hours 40 minutes). There are 6 limit orders to buy already in the books (2 at 3800 and 4 at 200), and 4 other limit orders to sell at 6,000.

7. The Spread • The spread is defined as the difference between the highest priced limit buy order ask price, called the bid price, and the lowest priced limit sell order called the ask. • In the example above, the ask price is 5,800 and the bid price is 3,800, so the spread is 2,000. • Observe that the trader whose display screen is illustrated reduced the spread from 2,200 by placing an order inside the previous bid ask quotes

8. Market orders are executed immediately • A market order to buy (sell) one unit is defined by a price which is greater (less) than or equal to the lowest (highest) outstanding limit order to sell (buy). • Therefore market orders transact instantaneously, market buy (sell) orders reducing the number of outstanding limit orders to sell (buy). • Market orders to sell (buy) are matched with the highest (lowest) priced limit order to buy (sell) and executed at the price of the matching limit buy (sell) order.

9. Limit orders are not always executed • All other orders (sell orders priced higher than the best bid price, buy orders priced lower than the best ask price) are entered in the book as limit orders. • A limit order can be withdrawn at any time before it is matched with an incoming market order as part of a transaction. • The outcomes of the random variables that determine asset valuation, the strategies of all the players, and the past history of the game, determine the execution probability of a limit order.

10. Solvency • To survive, a financial institution must enforce traders to honor contracts between themselves. • While the fiduciary rules vary across institutions, the free form module captures the essence of many, if not most. • In the market module the trader is constrained at each point in time by how much she can offer for sale within each market and how much she can buy in total. • This implies there are J constraints for placing sell orders but only 1 constraint for placing buy orders.

11. Constraints on sell orders Denote the set of buy and sell prices by For convenience we assume there are no short sales. Therefore the total amount of limit sell orders cannot exceed the holdings of a trader. Let sjkt denote the quantity of the jth asset for sell orders at price pk at time t, and let xjkt denote her total holdings of the asset at that teime. We require:

12. Constraint on buy orders An overall budget constraint on buy orders prevents the trader from placing orders that exceed her money holdings. Suppose there are J assets, each of which can be traded for a liquid asset called money. Let bjkt denote the quantity of the jth asset demanded at price k at time t, and let mt denote her cash (liquid assets) at that time. We require:

13. Description of a typical trading game Consider the problem a typical trader faces in a limit order market. First we define what we mean by a player type. Traders are distinguished by the ways in which they are constrained, the information they have and their preferences over asset allocations. Then we explain the trader’s optimization problem, and showing how the solution to that problem should be considered as part of a Nash equilibrium.

14. Player types • Subjects participating in a trading game experiment are assigned to be one of several player types. • Player types are characterized their: - market access - initial endowment - information - asset valuation

15. Market access • In the market module there is only one medium of exchange called money: all trades involve a transaction between money and a stock. (For example trading in more than one currency are excluded.) • A player might be allowed to submit limit and/or market orders to buy and/or sell in an asset market. • Depending on the game, money might be regarded purely as a medium of exchange, or stand for generalized purchasing power as well.

16. Which markets and how • At the beginning of a game each player is told what types of orders he is permitted to submit in each asset market. There are nine possibilities: • 1. Market buy • 2. Market sell • 3. Limit buy • 4. Limit sell • 5. Market buy and market sell • 6. Limit buy and limit sell • 7. Market buy and limit sell • 8. Market sell and limit buy • 9. Excluded from trading

17. Choices at time t • Players exploit their trading opportunities defined in the choice set to make decisions throughout the game. • At each successive instant t  [1, T] the trader may do nothing, or take an action in one market j  {1, . . . , J} subject to the constraints defined in the previous slides: • 1. Delete an existing limit order to buy or sell a quantity q in market j. • 2. Submit a market buy or a market sell order for quantity q. • 3. Submit a limit buy or a limit sell order for quantity q in market j at price pk.

18. Initial endowment • At the beginning of a game, players receive an initial endowment of: - the various stocks traded in the game - money

19. Information Players are also distinguished by the information : - they receive at the start of the game - how their information is updated throughout the game. Players might have different information about: - their own valuations - the valuations of others - the returns on the assets - how total quantity of assets held by the traders - the contents of the order book - the history of orders and transactions.

20. Valuation • In many market games we will play in this course the subjective valuation of a player for a unit of a given asset is its value to him/her if the game were to end instantly. • We denote by vjt the valuation of the trader for the asset j conditional on the game ending at time t. • This value is sometimes assumed to have two components: • a common value (to every player) • a private value which displaces each player’s subjective valuation from the common value.

21. Private value • The private valuation, is an individual specific component that is distributed according to a probability distribution for that player type. • When a subject logs on to the game, and is assigned to a player type, he receives an independent random draw from this distribution. • It represents the individual’s demand for a particular product/stock.

22. Common value • The common value for each stock is shared by every player. It is a (stochastic) process, which might represent the expected present value of discounted dividends streams. • The common value could be determined by factors, sometimes called its fundamentals. In this case the common value is a mapping from several factors, where the each factor follows a stochastic process. • Sometimes (not all) the traders know the common value. For example they might receive noisy signals about it.

23. Preferences We typically model preferences with a (concave increasing) utility function u(x) on consumption x where Then the overall objective that the trader uses to assess her trading and portfolio decisions is: or (if money has no value ):

24. Uncertainty When the jth asset is a stock, vjT is a random variable that represents the ex-post return on the value at the end of the game, It is therefore impossible to base trading decisions at time t < T on vjT because it is unknown at time t. Accordingly denote by lt the information available to the trader at time t. Assume that at each instant t the trader maximizes the expected value of the utility at T from her portfolio.

25. Expected value Then her expected utility may be expressed as: where: is the liquidation value of her stock holdings in the jth stock at time T.

26. The trader’s optimization problem • The trader sequentially makes the choices that at each successive instant t  [1, T] maximize: subject to the rules prescribing the orders he is permitted to place, the J budget constraints preventing short sales, and 1 overall budget constraint preventing borrowing.

27. The strategy space Let dj(lnt) be an indicator function showing which market the nth trader will choose to act in at time t when his information is lnt, where dj(lnt) =1 if he picks market j and dj(lnt) = 0 otherwise. Let q(lnt) denote the quantity he picks (where negative quantities indicate sell orders), and p(lnt) the price (which is constrained to be the market price if the trader is permitted to make only market orders are in this market). Then a strategy for the nth trader is the vector function s(lnt) = (dj(lnt), q(lnt), p(lnt)) for each t  [1, T].

28. The valuation function of trader n Let sn0(lnt)) denote the optimal order strategy, and define the value of correctly solving at time t by: Then for all t < r < T, the value function W(lnt) solves the recursion: This equation says that optimized value should behave like a random walk throughout the game.

29. Interdependence between players • The actions of the other players affect the trading opportunities of the nth player. • Consequently the probability distributions the nth player uses to take expectations over future events are partly determined by the trading strategies of the other players. • Therefore the previous two slides provide an incomplete description of the trader’s optimization problem, because they does not fully describe how to take the expectations over future trading opportunities.

30. The strategic solutionto limit order market game • Suppose each trader n = 1, . . . . ,N picks a strategy to solve their own optimization problem, and calculates the expectation knowing the strategy the other traders picked. • The resulting strategy choice se(lnt) for each trader n = 1, . . . . ,N is a symmetric Nash equilibrium for the limit order market game. It corresponds to the solution of the strategic form for this game (as defined in 45-974).

31. Trading in retail markets In a retail market stores set a sell limit order for multiple units of the good. Packages of goods are bundled by competing vendors. Buyers only submit buy market orders.

32. Trading in stock markets Consider some markets for financial assets on a stock exchange where participants make bids and offers at prices they choose throughout the duration of the game to maximize their expected subjective value of asset holdings at the end of the game. It might be convenient to imagine that the length of the game is the difference between the opening and closing time on a typical trading day.

33. Markets for Derivative Assets • When we buy or sell stocks we are trading the property rights to current ownership of an asset. • Markets are also created to trade the rights for future ownership, future ownership that is contingent on some future event, or ownership at the discretion of one or the other trader’s discretion.

34. Futures and Forward Markets • For example, a futures contract to sell a bushel of wheat is a legal obligation for the seller to provide a bushel of wheat at a designated time in the future for a price that is determined now. • A three month forward exchange rate is the amount you pay in domestic currency now to receive a unit of foreign currency in exactly three months. • Thus the price of a solvent foreign bond, the forward exchange rate and the current exchange rate (almost) matches up to the price of an equivalent domestic bond. Investors compare buying a domestic bond with buying foreign currency, selling foreign currency on the forward market and buying a foreign bond.

35. Markets for options • The owner of a call option on an asset unit entitles but does not obligate him to buy a unit of the asset at (or before) the end of the period for a previously agreed upon strike price. The person who wrote (supplied) the call is thus obligated to give up an asset unit in exchange for the strike price purely at the discretion of the trader who holds the call option. • Put options are defined symmetrically. Owning a put option entitles the trader, purely at his discretion, to exchange an asset unit with the player who wrote and sold it for the strike price.

36. Insurance Markets and debt • Insurance contracts are examples of contingent commodities. The policy holder pays a fixed fee when the period of coverage begins. • If the item is destroyed during the period of coverage, the firm who has written the insurance contract is obliged to replace the item. • Notice that writing (and marketing) forwards, options and forwards contracts on assets provides players with money now in exchange for future obligations. • For example writing a forwards contract on money is just a fancy name for seeking a loan.

37. Banks • Suppose investors have projects to undertake. • However they need cash to buy. • Banks provide the cash. • Sellers than then can transact with the buyers.

38. Trading in asset markets We turn to trading in financial assets. We discuss the objectives of traders, their choice variables variables that role of advance information, the speed with which traders can react, the size of the quotes traders set, and the number of stocks, the aggregates they manage and the manner in which they aggregate portfolios.

39. Auctions Most auctions are simple examples of limit order markets. The second half of this lecture describe the 5 main kinds of auctions. We conclude the lecture by analyzing how traders should bid in auctions, and how much revenue auctions generate when experienced bidders play their (equilibrium) strategies.

40. Why study auctions? • The study of auctions provides one way of approaching the question of price formation. • Auctions serve the dual purpose of eliciting preferences and allocating resources between competing uses. • A less fundamental but more practical reason for studying auctions is that the value of goods exchanged each year by auction is huge.

41. What kind of goods are sold by auction? • - Real estate (commercial, residential) • U. S. government uses auctions to sell Treasury bills; mineral, oil, and timber rights; and emissions permits; property rights over the frequency spectrum • agricultural produce • financial instruments • intangible properties (patents, trademarks) • procuring inputs from other firms • auctioning of airport time slots to competing airlines, proposed as an improvement over the existing slot quotas • privatization, takeover, merger, acquisition

42. Asymmetric information If the seller knows the valuations of the bidders, there might seem to be little reason for an auction: the seller could simply offer it to the bidder with the highest valuation at that price. Auctions are typically held when there is asymmetric information of various types: - a buyer knows more about his or her own valuation than the seller does - a buyer only receives a signal about how much he or she values the auctioned object.

43. Modeling valuations • In a private-value model each bidder knows how much she values the auctioned item, but her value is not known by the other players. • In a common-value model, all bidders would value the object the same way if they were fully informed, but bidders have different private information about the common value. • In a common-value model, a bidder would change her estimate of the value if she learnt another bidder’s signal, but in a private-value model her value is unaffected by learning any other bidder’s preferences. • In both models, however, learning the information of another player might affect bidding behavior.

44. Auction mechanisms There are 5 standard types of auctions for auctioning a single item which are widely used and analyzed: • First-price sealed-bid • Second-price sealed-bid • English • Japanese • Dutch as well as several other types we will investigate.

45. Sealed bid auctions • In sealed bid auctions each bidder in a sealed bid auction simultaneously submits a single price to the auctioneer, and the highest bidder receives the auctioned item. • Sealed bid auctions only differ in how much bidders pay. • We investigate three variations, first price, second price, and all pay.

46. First price sealed bid auction • A first price sealed bid auction can be defined in terms of limit and market orders. • Bidders simultaneously submit a single limit order to buy a unit without knowing any of the other bids, and then the auctioneer places a market order to sell one unit. • In a first price sealed bid auction, the winning bidder pays the amount she bid in exchange for the object up for auction. Therefore, un, the net payoff to the nth player is defined as:

47. Second price sealed bid auction • Each bidder in a second price sealed bid auction, also known as a Vickery auction, submits a single price to the auctioneer without knowing any of the other bids. • This auction cannot be defined in terms of limit and market orders. • The bidder submitting the highest price pays the second highest price submitted. The other bidders pay nothing and receive nothing. • The net payoff to the nth player is defined as:

48. Revenue comparison between the first and the second price sealed bid auction • Notice that b(²)≤b(¹) so that if all bidders adopted the same bidding strategy for both auctions, then the second price sealed bid auction would yield less revenue than the first price sealed bid auction, and the winners would pay less.

49. All pay sealed bid auctions • In an all-pay sealed bid auction, each bidder pays what she bids, and the highest bidder wins the auction. The net payoff to the nth player is defined as:

50. Examples of all pay auctions • All-pay auctions are a paradigm for modeling competitions of various kinds, not a common institution for literally conducting auctions. • For example supply contracts are like all-pay auctions. Bidders expend considerable resources preparing a proposal, but only one bidder is awarded the contract. • Similarly research teams in the same field use resources competing with each other, but the first team to make a discovery benefits disproportionately in the rewards from their discovery through patenting, first mover advantages, and so on.