UAI 2009 CONITZER 101 Prediction Markets, Mechanism Design, and Cooperative Game Theory Vincent Conitzer Dept. of Computer Science Duke University Durham, NC 27708, USA Abstract ket ensures that agents are rewarded for contributing useful and accurate information. Prediction markets are designed to elicit informa- To analyze prediction markets, it is often assumed that tion from multiple agents in order to predict (ob- agents will act myopically. In a market based on securi- tain probabilities for) future events. A good pre- ties, this corresponds to the assumption that an agent will diction market incentivizes agents to reveal their buy if the price is below her current subjective probability information truthfully; such incentive compat- (which takes what happened in the market so far into ac- ibility considerations are commonly studied in count), and will sell if the price is above her current subjec- mechanism design. While this relation between tive probability. Even under this assumption, it is theoreti- prediction markets and mechanism design is well cally possible that the market converges to a price that does understood at a high level, the models used in not reflect the full combined information of all the agents. prediction markets tend to be somewhat differ- For example, suppose that agent 1 observes x ∈ {0, 1}, ent from those used in mechanism design. This and agent 2 observes y ∈ {0, 1}. Let z = 1 if x = y, and paper considers a model for prediction markets z = 0 otherwise; and consider a prediction market that at- that fits more straightforwardly into the mecha- tempts to predict whether z = 1. In principle, the agents nism design framework. We consider a number collectively have enough information to predict z perfectly. of mechanisms within this model, all based on However, if we assume x and y are drawn independently proper scoring rules. We discuss basic properties according to a uniform distribution, then each agent’s sub- of these mechanisms, such as incentive compat- jective probability that z = 1 is 0.5 regardless of the in- ibility. We also draw connections between some formation she receives, so the price will remain stuck at of these mechanisms and cooperative game the- 0.5. (A similar example is given in [6].) Of course, this ory. Finally, we speculate how one might build is a knife-edge example. If we modify the distribution so a practical prediction market based on some of that P (y = 1) = 0.51, then, given an initial price of 0.5, these ideas. if agent 1 observes x = 1, she will start buying and drive the price up; whereas if she observes x = 0, she will start selling and drive the price down. From this behavior, agent 1 Introduction 2 can infer what agent 1 observed, and as a result knows z, so that the market price will correctly converge to 0 or 1. A prediction market [16] is a market created for the purpose A difficulty is presented by the fact that strategic agents of obtaining a subjective probability distribution, based on will not always behave myopically: they may try to manip- the information of multiple agents. To predict whether a ulate the beliefs of the other agents, and thereby, the market particular event (say, the Democratic candidate winning the price. Considering the modified example again, suppose election) will happen, a common approach is to create a se- that agent 1 changes her strategy to do the opposite of what curity that will pay out some predetermined amount (say, she did before. That is, she will start selling if x = 1, and $1) if the event happens, and let agents trade this security buying if x = 0. If agent 2 is not aware of agent 1’s strate- until a stable price emerges; the price can then (arguably) gic behavior, he will be misled into drawing the wrong con- be interpreted as the consensus probability that the event clusion about z, and drive the price to the exact opposite of will happen. However, there are also other designs for what it should be. This leaves agent 1 in an advantageous prediction markets. Examples include dynamic parimutuel position: if the price has been driven to 0, she can cheaply markets [14], as well as market scoring rules [10] (we will buy securities that in reality are worth 1; whereas if the discuss the latter in more detail). A good prediction mar- 102 CONITZER UAI 2009 price has been driven to 1, she can sell securities that are for the design of more practical markets: we could extend, worth 0 at a price of 1. A similar example is given in [13]. simplify, or otherwise modify parts of the general theory when this seems appropriate for the setting at hand.1 Also, We may ask ourselves whether we can avoid these difficul- towards the end of this paper, we will consider more prac- ties by designing the prediction market appropriately. The tical designs based on the ideas in this paper. creation of markets that lead to good results in the face of strategic behavior is a topic that falls under mechanism de- sign. A standard approach in mechanism design is to cre- 2 Background ate direct-revelation mechanisms, in which agents directly report all their private information, that are incentive com- In this section, we review the necessary background in patible, meaning that agents have no incentive to misreport proper scoring rules and market scoring rules, mechanism their private information. design, and cooperative game theory. The high-level idea that concepts from mechanism design should be applicable to prediction markets is certainly not 2.1 Proper scoring rules & market scoring rules novel. The relation is clear: in both cases, there are multi- ple agents with private information, and the goal is to in- Suppose our goal is to incentivize a single agent to truth- centivize them to reveal the (relevant) information. For fully report her subjective probability pE that an event E example, [11, 12] also study some versions of incentive will take place. We can do this by paying the agent some compatibility and other mechanism design concepts in the amount of money that depends both on the reported prob- context of prediction markets. Market scoring rules incen- ability pˆE, and on whether the event actually occurs. If tivize agents to report probabilities truthfully, if they are we let xE = 1 if the event occurs, and xE = 0 oth- myopic [13, 2]. erwise, then the agent receives a payment s(ˆpE, xE). s is said to be a proper scoring rule if the agent (uniquely) However, it appears that at this point, there is still some- maximizes her (subjective) expected payoff by giving her what of a gap between the theories of prediction markets true estimate of the probability—that is, for any p ∈ [0, 1], and mechanism design. This becomes especially appar- {p} = arg maxpˆ ps(ˆp, 1) + (1 − p)s(ˆp, 0). For simplic- ent when one considers, for contrast, how smoothly auc- ity, we will only consider settings with two outcomes—the tion theory integrates with mechanism design. The goal of event occurs, or it does not—but all of this is easily gen- this paper is to reduce (or perhaps eliminate) the gap, by eralized to settings with more outcomes (for example, by laying out a framework for prediction markets that fits bet- running a separate proper scoring rule for each outcome).2 ter with the theory of mechanism design. In particular, we Example proper scoring rules include the quadratic scoring 2 will consider a type of direct-revelation prediction market, rule, 1 − (xE − pˆE) [1], and the logarithmic scoring rule, in which agents report their full information directly (rather xE logp ˆE + (1 − xE) log(1 − pˆE) [8]. For the purpose of than trading securities, reporting probabilities, etc.). Like this paper, any proper scoring rule will do. market scoring rules, the framework is based on proper scoring rules. Interestingly, ideas from cooperative game Can a proper scoring rule be used in a setting with multiple theory also come into play (even though mechanism design agents? One elegant way of doing this is to use a market 3 is usually based on noncooperative game theory). We pro- scoring rule [10]. In a market scoring rule, there is a cur- rent estimate of the probability, pˆE. At any point in time, pose several specific mechanisms based on concepts such 0 any agent can change the current probability to any pˆE. In as the Shapley value and VCG mechanisms. 0 the end, this agent will be paid s(ˆpE, xE) − s(ˆpE, xE) for It should immediately be noted that direct-revelation pre- this change (which may be negative). This in some sense diction markets, strictly interpreted, are probably not prac- still gives the right incentive to the agent, because the agent tical. This is because agents in such a market need to be cannot affect s(ˆpE, xE). However, a market scoring rule able to directly reveal all their information that is pertinent to the prediction, including for example facts such as “I 1Even in auction theory, which interacts more fluidly with have interacted with some of the Democratic candidate’s mechanism design, the theoretical auction mechanisms from mechanism design (such as the generalized Vickrey auction) are staff and they seem very motivated and inspired.” It is generally not considered immediately practical, but they help es- hard to imagine a mechanism that can directly take such tablish a framework that is helpful in the design of more practical arbitrary natural language statements as input, determine a mechanisms, such as ascending auctions.