Strategic Betting for Competitive Agents

Transcription

1 Stategic Betting fo Competitive Agents Liad Wagman Depatment of Economics Duke Univesity Duham, NC, USA Vincent Conitze Depts. of Compute Science and Economics Duke Univesity Duham, NC, USA ABSTRACT In many multiagent settings, each agent s goal is to come out ahead of the othe agents on some metic, such as the cuency obtained by the agent. In such settings, it is not appopiate fo an agent to ty to maximize its expected scoe on the metic; athe, the agent should maximize its expected pobability of winning. In pinciple, given this objective, the game can be solved using game-theoetic techniques. Howeve, most games of inteest ae fa too lage and complex to solve exactly. To get some intuition as to what an optimal stategy in such games should look like, we intoduce a simplified game that captues some of thei key aspects, and solve it (and seveal vaiants) exactly. Specifically, the basic game that we study is the following: each agent i chooses a lottey ove nonnegative numbes whose expectation is equal to its budget b i. The agent with the highest ealized outcome wins (and agents only cae about winning). We show that thee is a unique symmetic equilibium when budgets ae equal. We poceed to study and solve extensions, including settings whee agents must obtain a minimum outcome to win; whee agents choose thei budgets (at a cost); and whee budgets ae pivate infomation. Categoies and Subject Desciptos I..11 [Distibuted Atificial Intelligence]: Multiagent systems; J.4 [Social and Behavioal Sciences]: Economics Geneal Tems Economics, Theoy Keywods Game theoy, Nash equilibium, stategic betting, contests, fai bets 1. INTRODUCTION Agents ae often evaluated accoding to some single-dimensional metic: fo example, the amount of cuency the agent has eaned, the numbe of points the agent has scoed, the numbe of tasks the agent has completed, etc. In settings with uncetainty, the design of the agent esults in a pobability distibution ove this metic. The agent s designe must optimize this distibution in some way. Most Cite as: Stategic Betting fo Competitive Agents, Liad Wagman and Vincent Conitze, Poc. of 7th Int. Conf. on Autonomous Agents and Multiagent Systems (AAMAS 8), Padgham, Pakes, Mülle and Pasons (eds.), May, 1-16., 8, Estoil, Potugal, pp. XXX-XXX. Copyight c 8, Intenational Foundation fo Autonomous Agents and Multiagent Systems ( All ights eseved. often, the designe will choose to maximize the expected scoe on the metic (expected amount of cuency eaned, etc.). Howeve, maximizing the expectation is not always the optimal thing to do. Most notably, agents ae often in competition with each othe, and the goal is to come out ahead of the othe agents. Fo example, in the Tading Agent Competition s Supply Chain Management game [7], compute pogams make decisions about managing a (simulated) supply chain. Thei tasks include negotiating supply contacts, bidding fo custome odes, and managing assembly and shipping. The winning agent is the one that has the most money in the bank at the end of the game. Anothe example is the AAAI Compute Poke Tounament [19], whee in the Bankoll Competition the winne is the agent with the most money in the end. Yet anothe example is the Penn-Lehman Automated Tading (PLAT) live competition, in which automated stock tading agents compete based on eal stock maket data [13]. (Unlike the othe competitions, this one is no longe active, though appaently not due to lack of inteest.) In games such as these, it can be tempting to ty to maximize one s expected amount of money, but in fact, the only thing that mattes is whethe the agent made moe money than the othe agents. Moe ecently, agent designes have stated to take this into account (fo example, [17]). As a simple numeical example, suppose that agent will cetainly end up with $5, and agent 1 has a choice between two stategies. Stategy 1 will give agent 1 $4 with pobability 1%; stategy will give agent 1 $6 with pobability 5%, and $1 with pobability 5%. The expected eanings of stategy ae $35, so if agent 1 aims to maximize expected eanings, it will choose stategy 1. Howeve, if the goal is to come out ahead of agent, stategy is the bette choice, since it esults in a 5% pobability of winning, wheeas stategy 1 esults in a guaanteed loss. Situations such as these, whee an agent has a choice between stategies that give oughly the same expected eanings but vey diffeent distibutions ove eanings, ae quite common fo example, the agent may be able to place vaious bets in (say) a casino, which will educe the agent s expected eanings only slightly but vastly incease the vaiance. It should be noted that this is not a citicism of maximizing expected utility. Rathe, it is a citicism of confusing eanings with utility. A sensible utility function hee would give utility 1 fo a win, and utility fo a loss. (Of couse, in some settings an agent may have some esidual utility fo money, so that the utility function consides both whethe the agent won and how much money the agent has. Howeve, at least in the competitions descibed above, the pedominant goal is simply to win.) Thee ae vey poweful axiomatic aguments fo maximizing expected utility (fo an oveview, see [16]), and nothing in this pape conflicts with maximizing expected utility.

2 Once the utility function is coectly defined, we can in pinciple solve such stategic settings using game-theoetic solution concepts, fo example, minimax stategies and Nash equilibia. Howeve, games such as the competitions mentioned above ae vey complex, and at least using cuent techniques, it is intactable to solve fo the game-theoetically optimal stategy although many ceative appoaches have been poposed to compute a stategy that is close to an optimal stategy, both fo the Tading Agent Competition [1,, ] and compute poke [18, 5, 11, 1]). We will make no attempt at solving them in this pape. Instead, we will study a much simple game that nevetheless illustates many of the key phenomena in these competitive settings. The most basic vesion of the game can be descibed as follows. Two agents, Alice and Bob, each have a budget of chips fo gambling. They each (simultaneously) place a single bet in (say) a casino. (We will assume that the outcomes of the bets ae independent.) Whoeve ends up with moe chips is named the winne, and chips ae wothless aftewads. What bets should Alice and Bob place? To answe this question, we need to know what bets the casino is willing to accept. Let us assume that, diven by competition, the casino is willing to accept any fai bet. 1 That is, an agent can buy any lottey (pobability distibution) ove nonnegative eal numbes whose expectation is equal to the agent s budget. Incidentally, if an agent wee able to place a sequence of bets, whee the choice of late bets is allowed to depend on the outcomes of the agent s own ealie bets (but not on the outcomes of the othe agent s bets), this would make no diffeence to the game, fo the following eason. Any plan (stategy) fo betting will esult in a (single) pobability distibution ove nonnegative numbes with expectation equal to the agent s budget, and thus the agent can simply choose this lottey as a single bet. In this pape, we study the equilibia of (the n-agent vesion of) this game, as well as vaiants in which agents must end up with at least a cetain numbe of chips to win; in which agents have to fist buy chips; and in which budgets ae pivate infomation. Thee is good eason to believe that the equilibium distibutions of these games bea some esemblance to the equilibium distibutions ove eanings in the agent competitions mentioned above. Fo example, in the stock tading competition mentioned above, (say) in the last day of tading, the agent can choose a potfolio that will esult in a paticula distibution ove eanings at the end of the day. The expected value of this potfolio at the end of the day will be oughly the same as its value at the beginning; howeve, the space of possible distibutions is vey lage, especially if it is possible to hold deivatives such as call and put options. 3 Again, the goal in the competition is simply to come out ahead of the othe agents. Because the equilibium distibutions in these competitions ae likely to be simila to those in the abstact game(s) in this pape, one can use ou esults in the following way: when ceating an agent fo one of these competitions, choose stategies that poduce appoximately the optimal distibution fo the game(s) studied in this pape. Indeed, the equilibium of ou game would suggest to 1 Real-wold casinos typically have payback ates of at least 9%. Unlike in casinos, in the stock maket iskie distibutions tend to have a slightly geate expected value. 3 One issue hee that is not modeled in this pape is that the values of the agents potfolios can be coelated, fo example because they hold the same stock, o because the values of diffeent stocks ae coelated (as they typically ae). Howeve, it is at least possible to ceate potfolios that ae oughly independent, fo example by investing in small companies fo which most of the isk is due to company-specific factos (divesifiable isk). hold quite isky potfolios in the stock tading competition which makes intuitive sense, as the goal is to come out ahead of the othes. While we have motivated ou esults fom a multiagent systems pespective, they ae also elevant to the study of seveal standad settings in economics. Fo example, pevious eseach in economics has consideed the stategic choice of lotteies as a means to chaacteize incentives fo isk-taking in R&D envionments. Hee, a choice of technology leads to a distibution ove the final quality (o impovement in quality) of the poduct, which detemines which fim will dominate the maket [1, 4, 6]. Patent aces constitute anothe application, whee again the choice of technology leads to a distibution ove the level of innovation, and the patent is awaded to the agent with the geatest innovation; howeve, hee, thee is typically also a minimum level of innovation that needs to be eached in ode to obtain the patent [8, 9]. (Late in the pape, we will study the vaiant of ou game whee agents must obtain at least a cetain value to win.) Othe applications include political campaigns and ams aces. In a woking pape, Dulleck et al. [1] (independently) popose what is effectively the same game as the basic setting that we initially study in this pape, in a diffeent context. They study all-pay auctions in which each bidde is budget constained, has no oppotunity cost fo thei budget, and has access to a fai insuance maket. (An all-pay auction is an auction in which each agent must pay its bid, even if it did not win. Fo an oveview on all-pay auctions, see [3]. Access to a fai insuance maket means that agents can place any fai bet.) Dulleck et al. ae motivated in pat by a esult by Laffont and Robet [15], who study the optimal (evenue maximizing) auction when biddes face (common knowledge) financial constaints. Laffont and Robet show that the optimal auction in this case takes the fom of an all-pay auction. Because of the equivalence of the games, all of ou esults also apply to this paticula type of all-pay auction. It must be admitted that this is not a vey common model of an all-pay auction (especially because biddes do not cae about how much money they have left in the end), and ou esults do not seem to have diect applications to moe common all-pay auction models. Dulleck et al. conside diffeent questions fom the ones in this pape, and consequently thei esults ae complementay to ous. They give an equilibium fo the case of two agents whose budgets ae not necessaily equal (ou Example ) and pove that this equilibium is unique. They also show that with n agents, an equilibium exists. In addition, they extend thei esults to allow fo multiple pizes a setting that we will not study in this pape. The emainde of ou pape is oganized as follows. In Section, we pesent the basic game and solve thee examples. In Section 3, we show that when agents have equal budgets, thee is a unique symmetic equilibium (which we povide explicitly). We exhibit some popeties of this equilibium, and we also show that unde cetain estictions on the lotteies, the symmetic equilibium is the unique equilibium of the equal-budget game. In Section 4, we extend ou symmetic equilibium chaacteization to the case whee agents must supass a minimum necessay outcome in ode to win. In Section 5, we study an extension of the basic game in which agents must fist select thei budgets (which come at a cost). In Section 6, we study an incomplete-infomation vaiant in which agents do not know the othe agents budgets.. THE BASIC GAME Let thee be n agents, and let agent i {1,..., n} be endowed with budget b i, which is common knowledge. (In Section 6, we extend the model to allow pivate budgets.) The basic game consists of two peiods. In the fist peiod, each agent (simultane-

3 ously) selects any fai lottey ove nonnegative eal numbes. 4 We descibe a lottey by its cumulative distibution function (CDF) F(x) : R [, 1]. That is, fo any x, F(x) is the pobability that the ealized lottey outcome is less than o equal to x. RAgent i s lottey F i is fai if its expectation is equal to b i, that is, xdf i(x) = b i. Thus, a pue stategy fo an agent in this game is any fai lottey ove nonnegative numbes. Any mixed stategy (consisting of a distibution ove lotteies a compound lottey in the [] famewok) can be educed to a pue stategy by consideing its educed lottey, the (simple) lottey that geneates the same ultimate distibution ove outcomes. Hence, we do not need to conside mixed stategies. (To eliminate any chance of confusion, because each distibution ove outcomes is a pue stategy, thee is no equiement that agents ae indiffeent among the outcomes in thei suppots in fact, natually, they will pefe the highe outcomes.) In the second peiod, each lottey s outcome is andomly selected accoding to its coesponding pobability distibution. The agent whose outcome is the highest wins. Fo now, we assume that agents only cae about winning. Thus, without loss of geneality, we assume that an agent gets utility 1 fo winning and fo not winning, so that the game is zeo-sum. (In Section 5, we extend the model to allow costly budgets.) Ties ae boken (unifomly) at andom. This gives ise to the following ex ante expected utility fo agent i: 5 U i(f i, F i) = R Q j i Fj(x)dFi(x). We will be inteested in the Nash equilibia F = (F1, F,..., Fn) of the simultaneous move game. Example 1. Conside the game between two agents, 1 and, with identical budgets b. Agent 1 s expected utility fom playing F 1 given that agent selects F is R F (x)df 1(x). Suppose that F is unifom ove [, b], so that F (x) = x/b fo x [, b] and F (x) = 1 fo x > b. Then, thee is no eason fo agent 1 to select a lottey that places positive pobability on outcomes stictly lage than b. This is because any pobability placed above b can be shifted down to b without loweing agent 1 s pobability of winning. Then, to make the lottey fai again, mass elsewhee can be shifted up, which can only impove agent i s expected utility. It follows that agent 1 s poblem is to select a distibution F 1 so R as to maximize 1 b xdf1(x) subject to the fainess condition b (hencefoth budget constaint) R b xdf1(x) = b. We note that the integal in the objective must equal b fo any F 1 that satisfies the budget constaint. Hence, any such F 1 constitutes a best-esponse to agent s stategy. Thus, it is an equilibium fo each agent to select the unifom lottey U[, b]. Moeove, because this is a twoagent zeo-sum game, lottey U[, b] is also a minimax stategy; it guaantees the agent an expected utility of at least 1/. This is in contast to the tivial stategy of just holding on to one s budget b, which can lead to an abitaily low expected utility: fo any ǫ (, 1), the opponent can put pobability ǫ on and pobability 1 ǫ on b/(1 ǫ), so that the opponent wins with pobability 1 ǫ. Example. Now, conside two agents with diffeent budgets, b 1 4 If negative lottey outcomes ae allowed, then an agent can place an infinitesimal mass on an extemely negative outcome, and distibute the est of its mass on lage positive outcomes. As a esult, no equilibium would exist. 5 Technically, the expession is only well-defined if the distibutions ae continuous, that is, they have no mass points. In a slight abuse of notation, we use the same expession fo distibutions with mass points (as is common in the liteatue). It should be noted that (fo example) in the two-agent case, if agent has a mass point at x, so that F (x) > lim ǫ F (x ǫ), then the pobability fo 1 of winning given that it obtains outcome x is not F (x), but athe lim ǫ F (x ǫ) + (F (x) lim ǫ F (x ǫ))/. This is only elevant if agent 1 also has a mass point at x. and b, and without loss of geneality suppose that b 1 < b. Suppose that agent s stategy F is the unifom lottey U[, b ]. Fist, we note that similaly to Example 1, thee is no eason fo agent 1 to select a lottey that places pobability on outcomes stictly lage than b. Thus, agent 1 s poblem is to select F 1 to maximize R b x b df 1(x) subject to R b xdf 1(x) = b 1. As befoe, any F 1 that satisfies the constaint constitutes a best-esponse fo agent 1. Now, conside the following compound lottey F 1: 1. Choose the lottey that with pobability b 1/b geneates outcome b, and with pobability 1 b 1/b geneates outcome.. If outcome b was geneated, then subsequently choose the lottey U[, b ]. Fomally, F 1(x) = 1 b 1/b + (b 1/b )(x/b ) ove [, b]. That is, agent 1 s lottey has a pobability mass at. (p is a mass point of a cumulative distibution function F if lim ǫ F(p + ǫ) F(p ǫ) >.) Lottey F 1 satisfies the constaint, and is thus a best esponse to F. Now, conside agent s poblem given that agent 1 uses F 1. With pobability 1 b 1/b, agent 1 gets (and given this, agent wins with pobability 1, as long as agent does not have a mass point at ), and with pobability b 1/b, agent faces the lottey U[, b ]. Since we have aleady detemined that U[, b ] is a best esponse against U[, b ], it follows that U[, b ] is a best esponse against F 1. Thus, we have found an equilibium. Again, because this is a two-agent zeo-sum game, the agents stategies ae also minimax stategies. Figue 1 shows the equilibium stategies gaphically. Cumulative Density 1 F 1 F b Outcome Figue 1: Equilibium stategies in Example Since agent 1 has a chance of winning only if it won its initial gamble, afte which it has the same budget as agent, its pobability of winning is b 1/b. We note that agent s equilibium stategy does not depend on b 1 (as long as b 1 b ). In contast, agent 1 s equilibium stategy does depend on b, because it places an initial, all-o-nothing gamble to even the odds and each b. [1] also study Examples 1 and, and show that the equilibium descibed hee is the unique equilibium in each case. Example 3. Now, suppose thee ae thee agents with identical budgets b, and conside the lottey F such that F(x) = (3b) 1 1 x ove [, 3b]. Given that agents and 3 employ stategy F, thee is no eason fo agent 1 to allocate mass to outcomes lage than 3b. Thus, agent 1 s poblem is to select F 1 to maximize R 3b F R (x)df 1(x) = 1 3b xdf1(x) subject to R 3b xdf1(x) = 3b b. As in Example 1, any lottey that satisfies the constaint is a best esponse. In paticula, playing F is a best esponse fo agent 1. Hence, (F, F, F) is a symmetic equilibium. In Section 3. we will illustate how symmetic equilibium stategies change as the numbe of agents inceases.

4 3. CHARACTERIZING EQUILIBRIA OF THE EQUAL-BUDGET GAME In this section, we will study the case whee all n agents have the same budget b >. We efe to this setting as the equal-budget game. We will show that this game has a unique symmetic equilibium. We also show that unde cetain conditions on the set of stategies, thee ae no othe equilibia. 3.1 Popeties of best esponses In this subsection, we pove that any best esponse in ou setting (even in games with unequal budgets) must have cetain popeties. These popeties will be useful in the emainde of this section, whee we analyze the equilibia of the equal-budget game. Conside agent i. Let F i(x) be the pobability that all agents othe than i obtain an outcome below x: F i(x) = Q j i Fj(x). The fist thee lemmas show that if i is best-esponding, then F i must be linea in the suppot of F i. (If this is not the case, then i is bette off changing its distibution, as we will show.) Fo given x 1 < x < x 3, Lemma 1 consides what happens if agent i shifts pobability fom (aound) x to x 1 and x 3, in an expectationpeseving way. If agent i is best-esponding, this cannot leave them bette off, and this imposes some constaints on F i. LEMMA 1. Conside x 1, x, x 3 R such that x 1 x x 3. Suppose that F i is continuous at x, and let F i be a best esponse fo i to F i. If x is in the suppot 6 of F i, then the following inequality holds: (x x 1)F i(x 3) + (x 3 x )F i(x 1) (x 3 x 1)F i(x ) Due to space constaint, we omit all the poofs; a full vesion is available upon equest. Nevetheless, to get some intuition fo why Lemma 1 is tue, suppose that F i has mass points at x 1, x, x 3. Suppose we modify F i by shifting ǫ mass fom x to x 1 and x 3. To peseve the expected value of the distibution, it must be that the mass shifted to x 1 is ǫ(x 3 x )/(x 3 x 1), and the mass shifted to x 3 is ǫ(x x 1)/(x 3 x 1). Since we assumed F i is a best esponse, this modification cannot have inceased the pobability that i wins. Hence, it must be that F i(x )ǫ F i(x 1)ǫ(x 3 x )/(x 3 x 1) + F i(x 3)ǫ(x x 1)/(x 3 x 1), which is equivalent to the expession in the Lemma. (The fomal poof addesses the geneal case whee F i does not necessaily have mass points.) Wheeas Lemma 1 consides shifting pobability mass fom outcome x to x 1 and x 3, Lemma consides the opposite. Intuitively, if outcomes x 1 and x 3 ae in the suppot of F i, then agent i should not find it pofitable to edistibute mass fom (aound) x 1 and x 3 to x in an expectation-peseving way. LEMMA. Conside x 1, x, x 3 R such that x 1 x x 3. Suppose that F i is continuous at x 1 and x 3, and let F i be a best esponse fo i to F i. If x 1 and x 3 ae in the suppot of F i, then the following inequality holds: (x x 1)F i(x 3) + (x 3 x )F i(x 1) (x 3 x 1)F i(x ) Lemma 3 follows immediately fom Lemmas 1 and, establishing that F i must be linea in the suppot of F i if i is best-esponding. LEMMA 3. Conside x 1, x, x 3 R such that x 1 x x 3. Suppose that F i is continuous at these outcomes and let F i be a best esponse fo i to F i. If x 1, x, and x 3 ae in the suppot of F i, then the following equality holds: (x x 1)F i(x 3) + (x 3 x )F i(x 1) = (x 3 x 1)F i(x ) Finally, we pove that the suppot of any best-esponse stategy has an uppe bound (unless the agent can win with pobability 1). 6 In ou use of the wod suppot, the suppot is a closed set, that is, we include all the limit points in the suppot. LEMMA 4. Given F i, suppose that thee is no stategy fo i such that i wins with pobability 1. Then the suppot of any best esponse stategy F i fo i has an uppe bound. The intuition behind Lemma 4 is the following. Shifting pobability mass that is placed on sufficiently lage outcomes downwads slightly will not decease the pobability of winning significantly. Doing so will allow the agent to shift mass on lowe outcomes upwads, whee this is moe fuitful. 3. Symmetic equilibia with equal budgets In the emainde of this section, we estict attention to the equalbudget game. Fist, in this subsection, we chaacteize the symmetic equilibia of this game. The esults we obtained in Subsection 3.1 assume that F i is continuous (at cetain points). The following lemma and coollay establish that in a symmetic equilibium, this assumption is tivially satisfied. LEMMA 5. Conside the equal-budget case. Suppose that the stategy pofile in which all agents play lottey F constitutes a (symmetic) equilibium. Then F has no mass points. Intuitively, if F had a mass point, then an agent would find it beneficial to deviate by shifting this mass up infinitesimally (to avoid a tie) and shifting mass down elsewhee. Since F is a cumulative distibution function with no mass points, F is continuous. F i is the poduct of continuous functions, and is thus continuous as well. We thus have the following coollay: COROLLARY 1. In the equal-budget game, suppose that the stategy pofile in which all agents play F constitutes a symmetic equilibium. Then F is continuous. Futhemoe, F i is continuous fo all i. We now show is in the suppot of any symmetic-equilibium stategy. LEMMA 6. Conside the equal-budget game. Suppose that the stategy pofile in which all agents play F constitutes a symmetic equilibium, and that the geatest lowe bound of the suppot of F is l. Then l =. To give some intuition, conside the following. If all agents playing F constitutes a symmetic equilibium and l >, then an agent s expected utility given that it obtained an outcome in a close neighbohood of l is nea. Hence, it is beneficial to eallocate mass in a neighbohood of l to and to some highe outcomes, contay to the equilibium assumption. We ae now eady to deive the main esult of this section. THEOREM 1. The equal-budget game has a unique symmetic equilibium. It is fo all agents to select the following lottey: F(x) = (nb) n 1 1 x n 1 1 (1) ove suppot [, nb]. If all agents use the lottey descibed in (1), then fo evey agent i, F i is the unifom distibution ove [, nb]. Hence, any lottey ove outcomes in [, nb] is a best esponse. Figue shows how the symmetic equilibium stategy changes with the numbe of agents. A andom vaiable that is of paticula inteest is the maximum outcome. This vaiable is especially inteesting when we intepet the game as a model fo competitive R&D, whee lotteies coespond to technologies that can be used and outcomes coespond to qualities of poducts. In this setting, the maximum outcome coesponds to the quality of the best poduct the one that will dominate the maket. The cumulative distibution of the maximum outcome in equilibium is (F(x)) n, and its expectation is: Z nb E[x max] = xd(f(x)) n = n b n 1 > nb

5 1 Cumulative Density b F F 3 F 4 b 3b 4b F 5 5b Outcome Figue : Cumulative distibution of symmetic equilibium stategy fo diffeent values of n, given equal budgets b = 5. This expectation is quite high, in the following sense. Suppose that we did not impose any stategic constaints on F i. Then, E[x max] E[ P i xi] = P i E[xi] = nb. That is, the expected value of the maximum outcome in equilibium is within a facto of the highest expectation that can be obtained without any equilibium constaint. (Incidentally, without the equilibium constaint one can in fact come abitaily close to achieving nb, as follows. Let F i be the distibution that places 1 ǫ mass on, and ǫ mass on b/ǫ. The pobability that at least one agent will eceive b/ǫ is 1 (1 ǫ) n, hence the expected quality of the poduct is (b/ǫ)(1 (1 ǫ) n ), which as ǫ conveges to nb.) Moeove, even if one can shift budgets among agents (in addition to pescibing thei stategies), it still holds that E[x max] nb. By contast, if each agent uses the degeneate stategy that places all the pobability mass on b, we would have E[x max] = b. 3.3 Uniqueness of the symmetic equilibium Is the symmetic equilibium unique, o do asymmetic equilibia exist? In this subsection, we show that unde mild estictions on the stategy space, the fome is the case. (We cuently do not know whethe these estictions ae necessay fo this to be tue.) Specifically, we conside the following estictions: (A1) Suppots have no gaps, (A) F i has no mass points fo all i {1,..., n}. The next lemma shows that if (A1) holds, then all agents have in thei suppot. LEMMA 7. Suppose that F = (F1, F,..., Fn) is an equilibium stategy pofile of the equal-budget game and that (A1) is satisfied. Then is in the suppot of Fi fo all i {1,,..., n}. We ae now eady to pesent the main esult of this subsection. THEOREM. Given (A1) and (A), the unique equilibium of the equal-budget game is the symmetic equilibium descibed in Theoem EXTENSION: MINIMUM OUTCOME REQUIREMENT In this section, we add one featue to the equal-budget game fom the pevious section: in ode to win, agents must end up with an outcome that is at least as high as some theshold. In othe wods, the winning agent must obtain the highest outcome among all agents, as well as each o exceed some minimum outcome. If no agent eaches this theshold, then no agent eceives anything. (We note that the game is no longe zeo-sum.) Let us denote this theshold by, whee >. Fo example, in a stock tading competition, thee may a specification that if a contestant does not outpefom a isk-fee asset, then the contestant cannot win. Similaly, in a tading agent competition, thee may be a specification that if no agent has positive pofit (which does sometimes happen: fo example, in the ealy ounds of the 3 Supply Chain Management TAC, as descibed in [14]), then nobody wins. Also, in the patent ace application fom economics, a minimum level of innovation must be eached o supassed fo a patent to be ganted. We wish to solve fo the symmetic equilibium of this modified equal-budget game. We will make use of the following obsevations. Fist, it is neve in agents inteest to select lotteies that place mass on outcomes in (, ). This is because outcomes in this inteval can neve lead to winning, so an agent would always be bette off eallocating mass fom this inteval to and to outcomes lage than. Second, Lemmas 3, 4, and 6 still hold in this context. Moeove, Lemma 3 can be extended to hold at even when F i is discontinuous thee, because outcomes close to can neve lead to winning when >. (We call this the "extended" Lemma 3.) Thid, Lemma 5 also holds, but only ove outcomes that ae at o above. Agents may have a mass point at. 4.1 The two-agent equal-budget game with a minimum necessay outcome Let us begin by solving fo the symmetic equilibium of the two-agent equal-budget game. By the above discussion, fo some h, the suppot of the symmetic stategy will be contained in {} [, h]. (Let h be the smallest numbe fo which this holds.) The next lemma shows that must be in the suppot. LEMMA 8. Conside the equal-budget game with a minimum necessay outcome of. Suppose that the stategy pofile in which all agents play F constitutes a symmetic equilibium. Let S denote the suppot of F, and let l be the geatest lowe bound of S {}. Then l =. Intuitively, the eason fo this esult is as follows. Suppose l >. Then, outcomes in a close neighbohood of l have a significant chance of leading an agent to winning only if all othe agents obtain outcome. Because of this, outcome povides almost the same pobability of winning as these outcomes. Thus, shifting mass fom a neighbohood of l to does not have a lage impact on an agent s pobability of winning, while it allows the agent to shift some mass to highe outcomes. Fo sufficiently small neighbohoods of l, doing so inceases the agent s pobability of winning. Theefoe, must be the geatest lowe bound of S {}. Lemmas 3, 5, and 8 imply that any symmetic equilibium stategy has the fom F(x) = a + cx ove [, h], whee a and c ae positive constants. Futhemoe, this stategy may place a mass m > at (so that F() = m). The following claim establishes that fo x [, h], F(x) must lie on a line oiginating fom the oigin. CLAIM 1. In the two-agent equal-budget game with a minimum necessay outcome of, thee is some c so that fo x [, h], F(x) = cx. (That is, a =.) Since F() = m, it holds that m = c. In addition, since F(h) = 1, we have that h = c 1. Finally, the budget constaint equies R c 1 xdf(x) = b. Substituting fo F in the constaint and eaanging, we obtain c(b, ) = + b b. Thus, the unique candidate symmetic equilibium stategy is fo each agent to select the lottey specified by 8 b + b if x < >< b F(x) = + b x if x () b + b >: 1 if x > b + b

6 It emains to veify that () indeed constitutes an equilibium stategy. To check this, suppose agent 1 employs stategy F. Given this, agent would not find it optimal to place mass on outcomes highe than c(b, ) 1. Thus, agent s poblem is to choose lottey F to maximize R c(b,) 1 F(x)dF (x) = c(b, ) R c(b,) 1 xdf (x) subject to R c(b,) 1 xdf (x) = b. Fo any F that satisfies the constaint and places no mass on (, ), R c(b,) 1 xdf (x) equals b, so the objective becomes c(b, ) b. Hence, any such F is a best esponse, including F. Figue 3 shows how the symmetic equilibium stategy vaies as inceases Cumulative Density = =5 =1 = Outcome Figue 3: Cumulative distibution of symmetic equilibium stategies fo diffeent values of, given equal budgets b = 5. We can obseve the following facts about the equilibium stategies fom () and Figue 3. Fist, as appoaches, c 1 (b, ) appoaches b, so that we convege to the equilibium of Example 1. Second, c(b, ) is deceasing in, so that, as gows lage, the cumulative distibution of the lottey chosen ove outcomes lage than becomes flatte. Meanwhile, the mass m at appoaches 1. Thus, the equilibium stategy becomes eve iskie as inceases. 4. The n-agent equal-budget game with a minimum necessay outcome We now extend the equilibium esult to n agents. THEOREM 3. In the n-agent equal-budget game with a minimum necessay outcome of, the unique symmetic equilibium stategy is fo each agent to play F descibed by 8 < m(b, ) if x < F(x) = (c(b, )x) n 1 1 if x [, (c(b, )) : 1 ] 1 if x > (c(b, )) 1 whee m(b, ) = (c(b, )) n 1 1 and c(b, ) is implicity and uniquely defined by 1 n (c 1 c n 1 1 n 1 n ) = b. As in the two-agent game, it can be veified that c(b, ) is inceasing in. Also, as appoaches, c(b, ) appoaches 1/nb, so that F becomes the unique symmetic equilibium stategy descibed in Theoem 1. Figue 4 shows how the symmetic equilibium stategy changes as n inceases. Figue 4 esembles Figue (whee thee is no minimum outcome equiement). One additional effect that the minimum outcome equiement intoduces is that as n gets lage, the mass that the equilibium stategy places on inceases in fact, this mass conveges to 1 as n. 5. EXTENSION: COSTLY BUDGETS In this section, we study a vaiant in which agents can choose thei budgets at the beginning of the game, and each budget comes Cumulative Density 1 F F F F Outcome Figue 4: Cumulative distibution of symmetic equilibium stategies fo diffeent values of n, given equal budgets b = 5 and = 1. at a cost. Afte the budgets have been chosen, the game poceeds as befoe. Thus, in the fist peiod, agents choose thei budgets b i; in the second peiod, they choose thei lotteies F i (whose expectation must equal b i); and in the thid peiod, outcomes ae dawn fom the lotteies and the winne is detemined. An agent s utility is b i if it does not win, and D b i if it does win, whee D is a constant. Agents ty to maximize expected utility. This vaiant is especially natual in many of the applications in economics, whee agents must make some initial investment. We only conside the -agent case, and we also do not conside the possibility of a minimum necessay outcome. To solve this game, we apply backwad induction. Suppose agent i has chosen budget b i in the fist peiod. To solve the subgame stating at the second peiod, we make use of the equilibium deived in Example (which, by the wok of Dulleck et al. [1], is unique). Assume without loss of geneality that b 1 b. (Even though the game is symmetic at the beginning, the agents may choose diffeent budgets in the fist peiod.) Fom Example, we know that it is an equilibium fo agent 1 to select lottey F 1(x) = 1 b 1/b +(b 1/b )(x/b ) and fo agent to select lottey F (x) = x/b, both with suppots [, b ]. (In fact, these ae minimax stategies.) Given this, we can analyze the fist peiod. Since the game is symmetic between agents at this point, it will suffice to focus on agent 1. Given that agent has decided on budget b >, agent 1 s expected utility as a function of b 1 is given by E[u 1(b 1, b )] = ( b 1 b D b 1 if b 1 b (1 b b 1 )D b 1 if b 1 > b When b 1 b, agent 1 s expected utility is linea in b 1. Hence, it will choose to set b 1 b wheneve D > b. Futhemoe, by diffeentiating the expected utility function when b 1 > b, it can be shown that b 1 = p b D/ maximizes expected utility, given that D > b. (We note that in this case, indeed, b 1 = p b D/ > b.) Moeove, it will choose to set b 1 = wheneve D < b, because in this case, any othe budget will give it a negative expected utility. Finally, when D = b, any b 1 [, D/] is optimal. To summaize, agent 1 s (set-valued) best-esponse function is 8 {} if b > >< D b 1(b ) = [, D ] if b = D >: q { b D } if < b < D We note that if b =, agent 1 would want to choose an infinitesimally small budget in ode to win, so the best esponse is not well-defined in this case. Figue 5 shows the agents best-esponse

7 cuves. (To eliminate any chance of confusion, we note that the D/ b D/ b1(b) b(b1) Figue 5: Best-esponse cuves in budget selection stage vaiables on the axes of this gaph ae budgets, not pobabilities; this gaph is not intended to show mixed-stategy equilibia.) The best-esponse cuves intesect at (D/, D/). The unique subgame pefect pue-stategy equilibium of this game is thus fo both agents to choose a budget of D/ in the fist peiod, and select the unifom lottey ove [, D] in the second. Each agent s expected utility is in equilibium. This is eminiscent of the equilibium of a common-value sealed-bid all-pay auction, whee both agents choose thei bids unifomly at andom fom [, D] (whee D is the common value), leading to an expected utility of fo each agent. We emphasize that while the equilibia ae simila, the games ae quite diffeent. 6. EXTENSION: PRIVATE BUDGETS In this section, we conside an incomplete-infomation setting, whee agents do not know the othe agents budgets. We conside the n-agent case, but do not conside the possibility of a minimum necessay outcome o costly budgets. Suppose that fo evey j {1,..., n}, agent j s (nonnegative) budget is selected by Natue accoding to some commonly known pio, descibed by the CDF W j(b). Thus, this is a Bayesian game, and we will use Bayes-Nash equilibium as ou solution concept. Suppose that agent j i chooses lottey G j b when endowed with budget b, and conside agent i s poblem. Given b i, agent i selects lottey F to maximize Z Z Y... G j b j (x)df(x)dw 1(b 1)... j i...dw i 1(b i 1)dW i+1(b i+1)...dw n(b n) subject to R xdf(x) = b i. Since agent i s expected utility is bounded by 1, Fubini s Theoem allows us to change the ode of integation in the objective function, which is hence equivalent to Z h Z Z Y... G j b j (x)dw 1(b 1)... j i (3) i...dw i 1(b i 1)dW i+1(b i+1)...dw n(b n) df(x) Hee, the backeted expession in (3) gives the ex ante cumulative distibution ove the maximum outcome of all agents othe than i, evaluated at x. Hence, the backeted tem has a ole that is analogous to the ole of F i(x) ealie in the pape: wheeas befoe the uncetainty deived only fom the othe agents stategies, now it deives both fom the othe agents stategies and fom Natue s choice of thei budgets. In ode to use ou pevious techniques fo deiving equilibia, we would need this expession to be popotional to x. This is illustated by the following two examples b1 of pio distibutions and coesponding stategies that constitute symmetic equilibia: 1. Conside the two-agent game with identical pio W = U[, h] fo some h >. One equilibium is fo both agents to acquie the degeneate lottey at b when endowed with a budget b. (This is because given these stategies, the distibution ove the othe agent s outcome is unifom ove [, h], hence any stategy that uses only outcomes in [, h] is a best esponse.). Fo some b >, let b L = 1 b and bh = 3 b. In a two-agent game with an identical pio P(b i = b L) = 1 and P(bi = bh) = 1, i {1, }, the stategy that chooses U[, b] when bi = bl and U[b, b] when b i = b H, constitutes a symmetic equilibium. (This is because given these stategies, the distibution ove the othe agent s outcome is unifom ove [, b], hence any stategy that uses only outcomes in [, b] is a best esponse.) Moe geneally, a stategy pofile G = (G 1,..., G n ), fo which fo evey i {1,..., n} the backeted tem in (3) is popotional to x fo all x that ae used in i s suppots, constitutes an equilibium. This is because, as in the complete-infomation case, the objective function educes to the constaint fo evey agent. Hence, any stategy that satisfies the constaint is a best esponse, including that suggested by G. Fo example, if the pio ove all agents budgets is W, with expectation k, then a stategy G that satisfies Z nk G b (x)dw(b) = (nk) 1 n 1 x 1 n 1 (4) fo all x [, nk], constitutes a symmetic equilibium. In ode to obtain such a stategy, we need to be able to tansfom the pio distibution W into anothe distibution. Specifically, we need stategy G to map budgets in the suppot of the pio W to fai lotteies, so that the ensuing (expected) distibution ove outcomes is as in (4). Let us say that pio CDF W is tansfomable into anothe CDF J if thee exists a stategy G such that the ensuing distibution is J. The following theoem povides necessay conditions fo a pio W to be tansfomable into a CDF J. THEOREM 4. Conside a CDF W and a CDF J, with suppots contained in R. Suppose that W is tansfomable into J. Then fo any b in the suppot of W, the following two inequalities must hold: R 7 b xdw(x) R J 1 (W(b)) xdj(x), and R R xdw(x) b xdj(x). J 1 (W(b)) Specifically, conside the case whee the pio ove each agent s budget is W, with expectation k. In ode fo thee to exist a stategy G that satisfies R nk G b (x)dw(b) = (nk) n 1 1 x n 1 1 fo all x [, nk] (and hence constitutes a symmetic equilibium), Theoem 4 tells us that fo any budget b in the suppot of W, it is necessay that E W [x x b] k(w(b)) n 1 and E W [x x > b] k P n 1 j= (W(b))j. It is an open question whethe these conditions ae also sufficient fo the stategy to be tansfomable in the desied way. Howeve, the following theoem does povide a (moe limited) sufficient condition: THEOREM 5. Conside a -agent pivate-budget game in which both agents budgets ae distibuted accoding to a commonly known 7 If J has mass points, then J 1 (W(b)) is not necessaily defined. In this case, R J 1 (W(b)) xdj(x) should be intepeted to integate x only ove the lowest W(b) mass of J. Letting y be the point such that J(y) > W(b) and J(y ǫ) < W(b) fo all ǫ >, a moe pecise expession would be R y xdj(x) (J(y) W(b))y. The intepetation of R xdj(x) is simila. J 1 (W(b))

8 CDF W with expectation k. If the suppot of W is a subset of [k/, 3k/], then W is tansfomable into U[, k] (and hence a symmetic equilibium exists). Intuitively, if W s suppot is a subset of [k/, 3k/], then given any budget, an agent can choose a fai lottey ove outcomes k/ and 3k/. Since W has expectation k, choosing such lotteies esults in a mass of 1/ at each of these outcomes. The agent can subsequently select lottey U[, k] given outcome k/, and U[k, k] given outcome 3k/. The esulting distibution ove outcomes is U[, k]. 7. CONCLUSIONS In many multiagent settings, each agent s goal is to come out ahead of the othe agents on some metic, such as the cuency obtained by the agent. Examples include tading agent competitions, compute poke tounaments, stock tading competitions, etc. In such settings, it is not appopiate fo an agent to ty to maximize its expected scoe on the metic; athe, the agent should maximize its expected pobability of winning. In pinciple, given this objective, the game can be solved using game-theoetic techniques. Howeve, the games above ae fa too lage and complex to solve exactly. To get some intuition as to what an optimal stategy in such games should look like, we intoduced a simplified game that captues some of thei key aspects, and solved it (and seveal vaiants) exactly. We expect that the equilibia of the lage games will display some similaity to the equilibia obtained in this pape. Specifically, the basic game that we studied is the following: each agent i chooses a lottey ove nonnegative numbes whose expectation is equal to its budget b i. The agent with the highest ealized outcome wins (and agents only cae about winning). We began by solving a few examples. Then, we studied the case whee each agent has the same budget. We showed that thee is a unique symmetic equilibium, in which each agent chooses a lottey that andomizes ove a continuum of monetay outcomes. The expectation of the highest ealized outcome in this equilibium is within a facto of what could be obtained if all agents coopeated to maximize the expectation of the highest ealized outcome. We also showed that unde some estictions on the lotteies, the symmetic equilibium is the unique equilibium of the equal-budget game. We poceeded to study vaiants of the basic game. Fist, we extended ou symmetic equilibium chaacteization to the case whee agents must supass a minimum necessay outcome in ode to win. Next, we studied a game in which agents fist choose thei budgets, which come at a cost. We found the unique puestategy subgame pefect equilibium of this game, which gives the agents an expected utility of. Then, we intoduced an incompleteinfomation model in which agents do not know the othe agents budgets a common situation. We showed that ou completeinfomation techniques can be applied to this setting if it is possible to tansfom the pio ove budgets into the appopiate distibution ove outcomes. We gave a necessay condition as well as a (moe estictive) sufficient condition fo this to be possible. Futue eseach can take a numbe of specific technical diections. The most obvious diections ae to extend ou esults to the setting of unequal budgets, as well as to investigate whethe the symmetic equilibium is the unique equilibium of the equalbudget game (without any estictions on the lotteies). Anothe impotant diection is to conside lottey spaces that ae esticted (fo example, allowing only lotteies ove a discetized space), o extended with unfai lotteies. Even moe geneally, we can allow agents to choose lotteies that ae coelated with each othe. Yet anothe diection is to conside vesions of these games in which agents may obseve othe agents budgets ove time. 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