Decision Markets With Good Incentives

Transcription

1 Decision Markets With Good Incentives Yiling Chen, Ian Kash, Mike Ruberry and Victor Shnayder Harvard University Abstract. Decision and prediction markets are designed to determine the likelihood of future events; prediction markets predict what will happen, and decision markets predict the results of a choice, or what would happen. Both allow multiple participants to review and make predictions, and participants are typically scored for improving the accuracy of the market s prediction. Previous work has demonstrated prediction markets can reward accuracy improvements, as can a single participant informing a decision. We construct and characterize decision markets where all participants are scored for improving the market s accuracy. These markets require the decision maker always risk taking an action at random, and reducing this risk increases its potential loss. We also relate these decision markets to sets of prediction markets, demonstrating a correspondence between their perfect Bayesian equilibria. 1 Introduction To make an informed decision a decision maker needs to understand the likely consequences of their actions. Decision markets are designed to predict these consequences, scoring a set of participants for improving the accuracy of the market s prediction. Consider, for example, a project manager deciding between two developers, A and B. The manager prefers to hire the candidate more likely to complete their work on time. Hanson [11] suggested running a decision market to determine the likelihood each will finish on time, using the market s prediction to hire a developer. Unlike a standard prediction market, which allows participants to predict future events (like if a developer will finish their work on time) and rewards them for improving the accuracy of the market s prediction, this proposed decision market was composed of two conditional prediction markets. One to determine the likelihood developer A would finish on time, conditional on A being hired, and the second to determine the likelihood developer B would finish on time, conditional on B being hired. This market would allow the project manager to make an informed hiring decision, if it could accurately reflect its participants information. Previous work has demonstrated prediction markets can aggregate their participants information in certain cases, and their predictions are often more accurate than those of other forecasting approaches [1,2,18,3,8]. Unfortunately, when a prediction market is used for decision making, the decision s dependence on the market s final prediction creates an incentive for participants to mislead the

2 decision maker, preventing the market from producing an accurate prediction and correctly informing a decision [16,6]. Returning to Hanson s proposed decision market and our example, we d like the market to reward participants for improving the market s prediction. Since there are two conditional markets, however, and only condition will be realized (only one developer is hired), participants are only scored for improving the market s prediction for the hired developer. When a participant has improved the market s prediction significantly on one developer and not other, then, it has an incentive to convince the project manager to hire that developer, regardless of how poor an employee that developer may be. More concretely, if the market currently predicts developer A has an 60% and developer B a 80% chance of finishing their work on time, and a participant believes the correct likelihoods are 70% and 80%, respectively, they only improve the market s accuracy for developer A. If developer B is hired this participant will receive a score of zero, but if A is hired they expected to score for a 10% improvement. Instead of being honest, then, the participant can pretend B is incompetent, lowering the market s prediction for the likelihood B will finish on time to less than 70%, cause A to be hired instead, and enjoy the profits. We address this manipulation and construct and characterize decision markets that, in expectation, always score predictions for their accuracy. Markets with this property are myopically incentive compatible, since participants always maximize their score for a prediction by honestly revealing their private beliefs. Instead of a scoring rule, these markets use a decision scoring rule that accounts for the likelihood actions are taken when scoring predictions. When the decision maker risks taking an action at random, a decision scoring rule can amplify the scores of unlikely actions and reduce the scores of likely ones, making risk neutral participants indifferent to their affects on the decision. We show this willingness to take an action at random is a requirement for these markets, and reducing this risk increases the decision maker s potential loss. While we do not directly discuss information aggregation in decision markets, we demonstrate a correspondence between their perfect Bayesian equilibria and those of a set of prediction markets. The rest of the paper is organized as follows. Section 1.1 describes previous work on prediction and decision markets. Section 2 provides a formal description of prediction markets in our notation, and Section 3 describes our decision market model. Section 4 presents our construction and characterization results. Finally, Section 5 discusses further research challenges and concludes. 1.1 Related work Eliciting and aggregating information from experts have been extensively studied. Strictly proper scoring rules can truthfully elicit the (subjective) probability distribution of an uncertain event from a single risk-neutral expert [14,17,10]. Hanson [12,13] turned strictly proper scoring rules into a prediction market mechanism called market scoring rules (MSR), which are myopically incentive

3 compatible for a sequence of experts. Both proper scoring rules and market scoring rules elicit probability distributions of an event and require that the actual event outcome be verifiable. In contrast, we consider mechanisms that elicit a set of conditional probability distributions of an event and only the event outcome for the realized condition is verifiable. The work by Othman and Sandholm [16] and Chen and Kash [6] is the closest to ours. Othman and Sandholm [16] were the first to formally study the incentive problem with eliciting conditional probability distributions for decision making. They considered the natural decision rule max, which, as in the previous project manager example, selects the action that maximizes the decision maker s expected utility, and devised a scoring rule which, when paired with the max decision rule, incentivizes a single risk-neutral expert to predict the conditional likelihoods consistent with its belief. Chen and Kash [6] considered more general decision rules, including both deterministic and stochastic rules, and characterized all scoring rules that are myopic incentive compatible for a single risk-neutral expert when paired with a decision rule. However, unlike strictly proper scoring rules and market scoring rules, such scoring rule and decision rule pairs do not straightforwardly extend the myopic incentive compatibility of the scoring rule to the market setting when multiple experts sequentially predict the conditional likelihoods and a decision is made based on the final market predictions. This is the setting we consider in this paper. Researchers have recently studied manipulation in prediction markets when there exist outside incentives, either in the form of profits in a related market [9] or some direct payoff depending on the final market prediction [5]. In this paper, we do not consider participants payoffs outside of the decision market. The decision maker s action may affect a participant s payoff within the market, but it doesn t change its utility outside of the market. 2 Prediction Markets: Background and Notation This section presents the standard market scoring rule model of a prediction market, first described by Hanson [12,13], and defines our notation. Let O be a finite, mutually exclusive, and exhaustive set of outcomes. A prediction market is a sequential game played by any number of risk-neutral, expected-value maximizing experts predicting the likelihood of these outcomes. The market opens at round zero with some initial prediction p 0 (O), where (O) is the set of probability distribution over outcomes. At each round after the market opens, an arbitrarily chosen expert makes a prediction p (O), and we let p t be the prediction made in round t. The market closes at some round t, after which an outcome o is observed and experts are scored for each prediction by a scoring rule, s : O (O) R { }, where R is the set of real numbers. We write s o (p) s(o, p) as a shorthand, and an expert s payment for a prediction is the difference between the scores of its

4 and the immediately preceding prediction. Letting T be the set of rounds when an expert made a prediction, its total payoff is s o (p t ) s o (p t 1 ). t T Markets with this sequential difference payoff structure are described as market scoring rule markets. Scoring rules are regular when only predictions assigning zero likelihood to the observed event are scored negative infinity, and proper when a risk-neutral expert s expected score for a prediction is maximized when predicting consistent with its beliefs, i.e. when it is myopic incentive compatibility. Formally, a rule is proper if for all beliefs q (O) over the likelihood of outcomes and predictions p q o s o (p), q o s o (q) o O o O where q o is the believed likelihood of outcome o. A rule is strictly proper when the inequality is strict unless q = p, uniquely maximizing an expert s score when they predict consistent with their beliefs. An example of a strictly proper scoring rule is s o (p) = a o + b log p o with a o R and b > 0. When a prediction market uses a strictly proper scoring rule, we say that it is strictly myopic incentive compatible. In aggregate, experts receive a payoff of Σ t t=1 s o (p t ) s o (p t 1 ) = s (p t o ) s o (p 0 ), so the market institution s worst-case loss is max s o (p 0 ) s (p t o ). o,p t Note that the market institution s payment is bounded and independent of the number of experts. In practice, a market institution s budget must be at least their worst-case loss. Ideally, the final prediction is an accurate consensus of experts beliefs. Bayesian experts, for example, update their beliefs as they observe other s predictions. However, while a market using a strictly proper scoring rule is myopic incentive compatible, it is not incentive compatible in general. An expert participating in multiple rounds may provide a prediction inconsistent with its belief, with the intention to mislead other experts and later capitalize on their mistakes [4]. Despite such possible manipulations by forward-looking Bayesian experts, previous work has shown that under certain conditions prediction markets that are myopic incentive compatible can fully aggregate information in finite rounds [4] or in the limit [15]. In this paper, when considering myopic incentive compatibility of decision markets, we do not restrict experts to be Bayesian but allow arbitrary beliefs. 3 Decision Market Model A prediction market is a special case of a decision market. Both use the same sequential market structure, but a decision market uses a decision rule to pick

5 from a set of actions before the outcome is observed, and which action is chosen may affect the likelihood an outcome occurs. Unlike previously proposed models of decision markets, we score experts using a decision scoring rule instead of a standard scoring rule. This more general function is necessary to recreate the myopic incentive compatibility of a prediction market for the broadest possible class of decision markets. Let A be a finite set of actions, and O a set of outcomes as before. Without loss of generality and for notational convenience we assume the outcomes for every action are the same. As in a prediction market, a decision market opens with an initial prediction, but instead of a single probability distribution it is a set of conditional distributions, one for each action, denoted P 0 (O) A. Experts still report sequentially, and we let P t be the prediction made in round t, P t a that prediction s distribution over outcomes given action a is chosen, and P t a,o be that conditional distribution s likelihood for outcome o. After the market closes, the decision maker selects an action using a decision rule D : (O) A (A), and the final report P t, drawing an action a from A according to the distribution D(P t ). We say that a decision rule has full support if it only maps to distributions with full support. As a shorthand we write d for a distribution over actions and d a the likelihood action a is drawn from the set. Once the action is selected, an outcome o is revealed, and experts are scored for each prediction by a decision scoring rule S : A O (A) (O) A R { }, written S a,o (d, P ) S(a, o, d, P ). Paralleling scoring rules, we describe decision scoring rules as regular when only predictions assigning zero likelihood to the observed event are scored negative infinity. Letting T again be rounds where an expert made a prediction, its total payoff is S a,o (d, P t ) S a,o (d, P t 1 ), t T so the market institution s worst-case loss is max S a,o (d, P 0 ) S a,o (d, P t ), (1) P t,a Ā,o O where Ā is the support of D(P t ). Previous work on decision markets used a similar model, but with a conditional scoring rule s c : A O (O) A R { }, instead of a decision scoring rule. As we show in the next section, however, considering the likelihood an action is selected is necessary to create the same myopic incentive compatibility as in prediction markets.

6 4 Decision Market Incentives In a prediction market, a strictly proper scoring rule uniquely maximizes an expert s score for a prediction when they predict consistent with their beliefs. The same is not always true in a decision market. While both markets can reward improvements over a prior prediction, a decision market only observes and scores the improvement in the prediction for the chosen action. Since this action is a function of the market s final prediction, experts may have an incentive to change this prediction (either directly or by manipulating other experts) to create a distribution over actions more likely to score their greatest improvement. In this section we extend the myopic incentives of prediction markets to decision markets, demonstrating how to construct myopic incentive compatible decision markets, and characterizing some of their properties. While myopic incentive compatibility does not guarantee that an expert who participates in multiple rounds will predict consistent with its beliefs in every round, previous work has shown that myopic incentives are sufficient to aggregate experts private information at perfect Bayesian equilibria under certain conditions [4,15]. Although we do not directly discuss information aggregation in decision markets, we establish a correspondence between the perfect Bayesian equilibria of any strictly proper decision market and a corresponding strictly proper prediction market. 4.1 Myopic Incentive Compatibility We first provide a formal treatment of myopic incentive compatibility for decision markets. Recall, for a prediction market, myopic incentive compatibility requires an expert always maximize their score for a prediction when they predict consistent with their beliefs. Assume a decision market uses decision rule D and decision scoring rule S. Then an expert with beliefs Q over the conditional outcomes who expects that d will be the final distribution over actions has an expected score for a prediction P of d a Q a,o S a,o (d, P ). a A o O And a myopic incentive compatible decision market must account not only for an expert s prediction, but also the likelihood each action is taken. Definition 1. A decision market (D, S) with a regular decision scoring rule S is proper if d a a A o O Q a,o S a,o (d, Q) a A Q a,o S a,o (d, P ), d a o O for all beliefs Q, distributions d and d in the codomain of D and predictions P. The market is strictly proper if the inequality is strict unless P = Q.

7 If a decision market is not strictly proper there exist final predictions and beliefs such that experts maximize their score for a prediction by misrepresenting their beliefs. Interestingly, the next result shows how to construct a strictly proper decision market given any decision rule with full support. Theorem 1. Let D be a decision rule with full support. Then there exists a decision scoring rule S such that (D, S) is strictly proper. Proof. The proof is by construction. Let s be any strictly proper scoring rule. Construct S a,o (d, P ) = 1 d a s o (P a ). (2) (D, S) is strictly proper: an expert s expected score for a prediction is 1 d a Q a,o s o (P a ) = Q a,o s o (P a ), d a a A o O a A o O the sum of the expected scores of the same prediction in a set of prediction markets, one for each action, using a strictly proper scoring rule. Since predicting consistent with beliefs maximizes the expected score of the expert in each market, it maximizes the sum of the expected scores. The construction in Theorem 1 requires a decision rule have full support, and makes experts expected scores independent of future reports while their actual scores vary inversely with the likelihood an action is chosen. Surprisingly, every strictly proper decision market with a differentiable decision scoring rule has these properties. We prove the necessity of full support before characterizing all strictly proper decision market with differentiable decision scoring rules. Theorem 2. Let D be a decision rule. A decision scoring rule S that makes (D, S) strictly proper exists if and only if D has full support. Proof. First we prove that if a decision rule D does not have full support, there is no decision scoring rule S such that (D, S) is strictly proper. We proceed by contradiction. Let D be a decision rule without full support, choose a final report P so that d = D(P ) has d k = 0 for some k A, and let S be a decision scoring rule such that (D, S) is strictly proper. Let Q, Q (O) A be two beliefs differing only on action k: for all a k and all o, Q a,o = Q a,o; o Q k,o Q k,o. Consider the expected utility of an expert with each of these beliefs reporting truthfully, while the final report remains P. One of these utilities must be weakly greater than the other. Without loss of generality, let d a Q a,os a,o (d, Q ), (3) a A o O d a Q a,o S a,o (d, Q) a A Because Q and Q only differ on action k, and d k = 0, d a Q a,os a,o (d, Q), (4) a A o O d a Q a,o S a,o (d, Q) = a A o O o O

8 Combining lines (3) and (4) contradicts strict properness with respect to Q. The other direction, which shows how to construct a strictly proper decision market for any decision rule with full support, follows by the construction in the proof of Theorem 1. Theorem 2 gives a positive answer to Chen and Kash s open question about whether it is possible to construct decision markets with good incentives [6]. It also extends Othman and Sandholm s impossibility result for deterministic decision markets [16] to the more general class of decision markets without full support. The theorem does not apply to non-strictly proper decision markets; for example, all constant decision scoring rules are proper for all decision rules. Theorem 3 characterizes proper and strictly proper decision markets that use decision rules with full support, and parallels similar proper scoring rule characterization theorems given by Gneiting and Raftery [10] and Chen and Kash [6]. Theorem 3. A decision market (D, S), where S is regular and D has full support, is (strictly) proper if S a,o (d, P ) = 1 ( G(P ) G (P ) : P + A G d a A a,o(p ) ) (5) where G : (O) A R is a (strictly) convex function, G (P ) is a subgradient of G at P and : denotes the Frobenius inner product. Conversely, if S is differentiable in P and (D, S) is (strictly) proper, then S can be written in the form of (5) for some (strictly) convex G. The proof appears in Appendix A, and uses techniques similar to those in Gneiting and Raftery [10] and Chen and Kash [6]. Like the construction of Theorem 1, the characterization shown in Theorem 3 requires an expert s expected score to be independent of the final report, and that the realized score vary inversely with the likelihood an action is taken. It does, however, allow more complicated constructions than the normalized strictly proper scoring rules used in Theorem 1. For example, given a decision rule D with full support, defining S a,o (d, P ) = 1 d a A (2 A P a,o i,j P 2 i,j), (6) makes (D, S) a strictly proper decision market. Theorem 3 also illustrates that our expansion of the payment rule in decision markets from scoring rules to decision scoring rules is necessary to obtain myopic incentive compatibility, because scoring rules do not allow a dependence on d. 4.2 Approximating Deterministic Decisions Deterministic decision rules, like max, are natural: given the decision maker s beliefs after the market closes, deterministically choose the action maximizing

9 expected utility. Unfortunately, no strictly proper decision market can use a deterministic decision rule. However, it is possible to approximate deterministic decision rules with stochastic ones, but better approximations of a deterministic decision rule increase the decision maker s worst case loss. Corollary 1. Every strictly proper decision market (D, S) where inf P (O) A D a (P ) = 0 for some action a has unbounded worst-case loss. We omit the proof as it follows directly from the inverse relationship between scores and the likelihood of actions required by Theorem 3. In practice, budget-constrained decision makers using a strictly proper decision market may want to maximize their expected utility. Without a budget constraint, the decision maker could approximate max arbitrarily closely. However, the budget constraint imposes a lower bound on the probability of any action being chosen. Given a budget constraint and decision scoring rule, the approx-max decision rule maximizes expected utility by assigning this lower bound to every suboptimal action. approx-max works as follows: given a budget b and a decision scoring rule S, compute the lower bound on the probability for each action a as p a = max P,o S a,o (d a, P ) S a,o (d a, P 0 ), b where d a is the vector which assigns probability one to a and zero to all other actions. 1 The correctness of this formula follows from the required relationship between d and S from Theorem 3. When given the final market report P t, compute the action a that maximizes expected utility with respect to the prediction in P t. Let the decision probabilities be d a = 1 a a p a, and d a = p a for a a. This is a well defined decision rule that can be published at the start of the market, as long as the decision maker s utilities are known, and maximizes the chance that the optimal action is chosen, while avoiding going over the budget even when low-probability actions are selected. Note that the bound is with respect to a given decision scoring rule. If the decision scoring rule were also allowed to vary, arbitrary approximations with any budget would be achievable by scaling all payoffs by a sufficiently small constant. In practice, however, if experts have extremely small expected utilities, they may decline to participate, which imposes a limit on the decision maker s ability to reduce payoffs. 4.3 A Correspondence Between Decision Markets and Prediction Markets As mentioned in Section 1.1, previous work on prediction markets has demonstrated that myopic incentives are sufficient to aggregate information for Bayesian experts in equilibrium under certain conditions [4,15]. While we do not extend 1 If this results in a pa > 1, the budget is not sufficient for the specified S, even with a uniform decision rule.

10 these results directly to decision markets, we show a correspondence between the perfect Bayesian equilibria of a strictly proper decision market and a set of strictly proper prediction markets, suggesting current aggregation results may generalize to decision markets, too. Intuitively, the correspondence follows from thinking of a decision market as a set of prediction markets, one for each action, each with the same set of outcomes for convenience. After the decision market s close, the decision maker uses its decision rule to select one prediction market to observe, and uses that outcome to score experts predictions. While in a set of prediction market every market s outcome is observed, the expected score for each prediction in both markets can be the same with an appropriate chosen decision scoring rule and corresponding multi-scoring rule for the set of prediction markets. Because every action must be taken with positive probability, the decision scoring rule increases the scores of unlikely actions and lowers the scores of likely ones, in expectation creating the same distribution as a set of prediction markets. Because both markets predict the same set of outcomes, and each prediction has the same expected score, it is implied that expected value maximizing experts behave the same in both cases, as Theorem 4 formalizes. Theorem 4. A set of strategy profiles and beliefs is a perfect Bayesian equilibrium of a strictly proper decision market using a regular, differentiable decision scoring rule, only if it is a perfect Bayesian equilibrium of a corresponding set of strictly proper prediction markets. Our Bayesian model, the correspondence, and a proof sketch appears in Appendix B. 5 Conclusion and Discussion We extended the myopic incentive compatibility of prediction markets to decision markets. We proved that this extension requires the decision maker use a decision rule with full support, and showed how to construct a strictly proper decision market for any such decision rule, answering an open question posed by Chen and Kash [6]. We characterized the set of myopic incentive compatible decision markets, and show that it is possible to approximate any deterministic decision rule with a stochastic decision rule, although better approximations cause higher worst-case loss for the decision maker. Finally, we showed a correspondence between strictly proper decision markets and a sets of strictly proper prediction markets, suggesting that previous results about information aggregation in prediction markets may extend to decision markets. The requirement that the decision maker commit to a randomized decision rule poses an important practical challenge for decision markets. Returning to our example from the introduction, the project manager must be willing to risk hiring the slower developer to run a strictly proper decision market. This is not credible behavior a manager prefers to hire the faster developer. Creating credible decision markets with good incentives is likely to be a prerequisite for the

11 deployment of decision markets in practice. Commitment devices may provide credibility. Alternatively, in some restricted settings, non-strictly proper decision markets may still have good incentives. Another practical concern is presenting decision markets to the experts. Prediction markets using market scoring rules are often presented as a market with tradable contracts whose payout only depends on the realized outcome, with the market price of each contract indicating the current prediction. Such markets are operated using a cost function derived from the market scoring rule and have identical expert incentives to the sequential report model we described, but may be more natural to interact with [7]. We can design tradable contracts for decision markets, but because of the necessary dependence on the final decision rule probabilities, the contracts have a more complex structure and require large up-front payments to ensure that experts never owe money after the market closes. Designing a simpler and more intuitive contract structure for experts in a decision market would be of considerable practical value. References 1. J. E. Berg, R. Forsythe, F. D. Nelson, and T. A. Rietz. Results from a dozen years of election futures markets research. In C. A. Plott and V. Smith, editors, Handbook of Experimental Economic Results J. E. Berg and T. A. Rietz. Prediction markets as decision support systems. Information Systems Frontier, 5:79 93, K.-Y. Chen and C. R. Plott. Information aggregation mechanisms: Concept, design and implementation for a sales forecasting problem. Working paper No. 1131, California Institute of Technology, Division of the Humanities and Social Sciences, Y. Chen, S. Dimitrov, R. Sami, D. M. Reeves, D. M. Pennock, R. D. Hanson, L. Fortnow, and R. Gonen. Gaming prediction markets: Equilibrium strategies with a market maker. Algorithmica, 58(4): , Y. Chen, X. A. Gao, R. Goldstein, and I. A. Kash. Market manipulation with outside incentives. In AAAI 11: Proceedings of the 25th Conference on Artificial Intelligence, Y. Chen and I. A. Kash. Information elicitation for decision making. In AAMAS 11: Proceedings of the 10th International Conference on Autonomous Agents and Multiagent Systems, Y. Chen and D. M. Pennock. A utility framework for bounded-loss market makers. In UAI 07: Proceedings of the 23rd Conference on Uncertainty in Artificial Intelligence, pages 49 56, S. Debnath, D. M. Pennock, C. L. Giles, and S. Lawrence. Information incorporation in online in-game sports betting markets. In EC 03: Proceedings of the 4th ACM conference on Electronic commerce, pages , New York, NY, USA, ACM. 9. S. Dimitrov and R. Sami. Composition of markets with conflicting incentives. In EC 10: Proceedings of the 11th ACM conference on Electronic commerce, pages 53 62, New York, NY, USA, ACM. 10. T. Gneiting and A. E. Raftery. Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association, 102(477): , 2007.

12 11. R. Hanson. Decision markets. IEEE Intelligent Systems, 14(3):16 19, R. D. Hanson. Combinatorial information market design. Information Systems Frontiers, 5(1): , R. D. Hanson. Logarithmic market scoring rules for modular combinatorial information aggregation. Journal of Prediction Markets, 1(1):1 15, J. McCarthy. Measures of the value of information. PNAS: Proceedings of the National Academy of Sciences of the United States of America, 42(9): , M. Ostrovsky. Information aggregation in dynamic markets with strategic traders. In EC 09: Proceedings of the tenth ACM conference on Electronic commerce, page 253, New York, NY, USA, ACM. 16. A. Othman and T. Sandholm. Decision rules and decision markets. In Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pages , L. J. Savage. Elicitation of personal probabilities and expectations. Journal of the American Statistical Association, 66(336): , J. Wolfers and E. Zitzewitz. Prediction markets. Journal of Economic Perspective, 18(2): , A Proof of Theorem 3 This appendix gives the proof of Theorem 3, restated here for convenience. Theorem 5. A decision market (D, S), where S is regular and D has full support, is (strictly) proper if S a,o (d, P ) = 1 ( G(P ) G (P ) : P + A G d a A a,o(p ) ) (7) where G : (O) A R is a (strictly) convex function, G (P ) is a subgradient of G at P and : denotes the Frobenius inner product. Conversely, if S is differentiable in P and (D, S) is (strictly) proper, then S can be written in the form of (7) for some (strictly) convex G. Proof. For simplicity, we state the proof only for strictly proper markets, as that is the more interesting case. To obtain the proof for proper markets, replace strictly convex with convex, and make all inequalities weak. First, we show that if D is a decision rule with full support, defining S as in (7) makes (D, S) a strictly proper decision market. Let G be a strictly convex function, and let S be a decision scoring rule defined using (7). Writing the expected score for S, with final decision probabilities d, expert beliefs Q, and expert report P, we have d a Q a,o S a,o (d, P ) a A o O = a A o O 1 ( d a Q a,o G(P ) G (P ) : P + A G d a A a,o(p ) ) = G(P ) + (Q P ) : G (P ),

13 The expert s expected score with a report of Q is G(Q) + (Q Q) : G (Q) = G(Q) and since we assumed G is strictly convex and G is its subgradient, for all Q and P Q G(Q) > G(P ) + (Q P ) : G (P ). This establishes strict properness because the expected score for reporting Q is strictly greater than the expected score for any deviation P. Now we prove the other direction. Given a strictly proper decision market (D, S), where S is regular and differentiable in P, we need to define G such that S is of the form specified by (7) and show that G is strictly convex. First, we define an expected score function for final distribution d, report P, and beliefs Q: V (d, P, Q) = d a Q a,o S a,o (d, P ). a A o O Note that our definition of a strictly proper decision market immediately implies that the expected score for truthful reporting must be independent of d: for any d and d and any Q, V (d, Q, Q) = V (d, Q, Q). (8) Now we define G. Letting d and d be any distributions in the range of D, we define G to be the expected score for a truthful report: G(P ) = V (d, P, P ) First, we show that G is convex. Because S is proper, G(P ) = sup P V (d, P, P ) and since V (d, P, P ) is convex in P, G(P ) is the pointwise supremum of a set of convex functions, and hence is convex itself. Since S is differentiable in P, G is differentiable and has a unique subgradient (the gradient). Next, we show that the gradient of G is G a,o(p ) = d as a,o (d, P ).

14 Consider beliefs Q and a report P Q: G(P ) + (Q P ) : G (P ) = V (d, P, P ) + (Q P ) : G (P ) = V (d, P, P ) + (Q a,o P a,o )d as a,o (d, P ) a A o O = V (d, P, P ) + V (d, P, Q) V (d, P, P ) = V (d, P, P ) + V (d, P, Q) V (d, P, P ) (9) = V (d, P, Q) < V (d, Q, Q) (10) = V (d, Q, Q) (11) = G(Q) Lines (9) and (11) follow by (8). The inequality in line (10) follows because we assumed that (D, S) was strictly proper. Because we have G(P ) + (Q P ) : G (P ) < G(Q) for all Q and P Q, G is the gradient of G. Note that we obtained this result without any restriction on d. Because the gradient is unique, this implies that the value of G does not depend on d, so d as a,o (d, P ) = d as a,o (d, P ), for any d. It remains to show that S is of the form given in (7). Letting d be any distribution, we have 1 ( G(P ) G d (P ) : P + A G a,o(p ) ) a A = 1 ( V (d, P, P ) V (d d, P, P ) + A G a,o(p ) ) a A = 1 d a A A G a,o(p ) (12) = 1 d a d as a,o (d, P ) = 1 d a d as a,o (d, P ) (13) = S a,o (d, P ) Line (12) again follows by (8), and line (13) follows by the the uniqueness of the gradient discussed earlier. This concludes the proof. B Our Bayesian model and proof of Theorem 4 In this appendix we present a brief Bayesian model and sketch a proofs of Theorem 4 using pure strategy profiles for simplicity of exposition. The markets function as described in Sections 2 and 3 respectively, but we explicitly model the process by which an outcome is selected and experts receive and update their

15 beliefs. We first describe a set of prediction markets, then a decision market, and conclude with the formal statements. Let a set of prediction markets have some number of individual markets m, each, for notational convenience, with the same outcomes O. Before the market opens, Nature picks a set of outcomes, denoted O, by drawing an element from O m according to a common prior probability distribution q. We let O 0... O m 1 be each market s respective chosen outcome. Each expert i then receives a private signal Q i (O) m from a commonly known private signal generating function Π(i, O ). The market opens with an initial prediction P 0 (O) m and in rounds experts predict one probability distribution per market. After the market closes, Nature reveals O and experts are scored by a multi-scoring rule, s : O (O ) R { }. Interestingly, the multi-scoring is also a scoring rule over the set of outcomes O. A multi-scoring rule is strictly proper if E[s (O, Q) Q] > E[s (O, P ) Q], where E[s (O, Q) Q] is the expected value for an expert reporting consistent with their beliefs, for all P Q. We also describe a set of prediction markets as strictly proper when they use a strictly proper multi-scoring rule. Theorem 3 identifies the following as a strictly proper differentiable multi-scoring rule m 1 s (O, P ) = G(P ) G (P ) : P + G i,o i (P ), (14) it being a special case where every action is taken with probability one. Decision markets operate like a set of prediction markets, except Nature only reveals a single outcome after the market s close instead an outcome per action. More precisely, let a strictly proper decision market have actions A, common outcomes O, decision rule D and decision scoring rule S. Then we can consider it a set of m = A prediction markets, each with outcomes O. Instead of revealing O as before, however, Nature draws from O according to D(P t ), the decision rule applied to the markets closing prediction, and reveals only the one outcome drawn, denoted o. Every strictly proper decision market with a differentiable decision scoring rule S has S a,o (d, P ) = 1 ( G(P ) G (P ) : P + A G d a A a,o(p ) for some strictly convex function G, according to Theorem 3. The expected value of such a function in any strictly proper decision market is 1 ( Q a,o G(P ) G (P ) : P + A G A a,o(p ) ) a A o O = G(P ) G (P ) : P + Q a,o G a,o(p ), a A o O i=0

16 the same as the expected value of a prediction in a set of m prediction markets with outcomes O using a multi-scoring rule of the form given in (14). This is the correspondence Theorem 4 exploits. Theorem 6. A set of strategy profiles and beliefs is a perfect Bayesian equilibrium of a strictly proper decision market using a regular, differentiable decision scoring rule, only if it is a perfect Bayesian equilibrium of a corresponding set of strictly proper prediction markets. We sketch the proof as follows for the special case where experts play pure strategies. Let the correspondence be as described above. Then, for an expert with the same beliefs the same prediction has the same expected value. Now consider a set of pure strategy profiles and private signals, and we ll show experts have the same expected value for this strategy profile in both games, implying the set of perfect Bayesian equilibia are the same in both, too. Proof by induction. Base case: the first expert has the same beliefs in both games by assumption, and therefore the same expected value for every prediction. Applying its pure strategy profile yields the same prediction in both games. Inductive step: assume the first k experts have made the same prediction in both games. Expert k + 1 updates their beliefs using Bayes rule conditional on the prior actions, which are the same in both games, so expert k+1 has the same beliefs in both games, therefore the same expected value for every prediction, and therefore it makes the same prediction. Since all predictions made in both games are the same, the beliefs are the same at each stage, and the expected value for each prediction given the same beliefs is the same in both games, experts have the same expected value for each set of strategy profiles and private signals.