Abstract. We study the equilibrium behavior of informed traders interacting with two types of automated market makers: market scoring rules (MSR) and dynamic parimutuel markets (DPM). Although both MSR and DPM subsidize trade to encourage information aggregation, and MSR is myopically incentive compatible, neither mechanism is incentive compatible in general. That is, there exist circumstances when traders can benefit by either hiding information (reticence) or lying about information (bluffing). We examine what information structures lead to straightforward play by traders, meaning that traders reveal all of their information truthfully as soon as they are able. Specifically, we analyze the behavior of risk-neutral traders with incomplete information playing in a finite-period dynamic game. We employ two different information structures for the logarithmic market scoring rule (LMSR): conditionally independent signals and conditionally dependent signals. When signals of traders are independent conditional on the state of the world, truthful betting is a Perfect Bayesian Equilibrium (PBE) for LMSR. However, when signals are conditionally dependent, there exist joint probability distributions on signals such that at a PBE in LMSR traders have an incentive to bet against their own information—strategically misleading other traders in order to later profit by correcting their errors. In DPM, we show that when traders anticipate sufficiently better-informed traders entering the market in the future, they have incentive to partially withhold their information by moving the market probability only partway toward their beliefs, or in some cases not participating in the market at all. 1

Abstract We study the equilibrium behavior of informed traders interacting with market scoring rule (MSR) market makers. One attractive feature of MSR is that it is myopically incentive compatible: it is optimal for traders to report their true beliefs about the likelihood of an event outcome provided that they ignore the impact of their reports on the profit they might garner from future trades. In this paper, we analyze non-myopic strategies and examine what information structures lead to truthful betting by traders. Specifically, we analyze the behavior of risk-neutral traders with incomplete information playing in a dynamic game. We consider finite-stage and infinite-stage game models. For each model, we study the logarithmic market scoring rule (LMSR) with two different information structures: conditionally independent signals and (unconditionally) independent signals. In the finite-stage model, when signals of traders are independent conditional on the state of the world, truthful betting is a Perfect Bayesian Equilibrium (PBE). Moreover, it is the unique Weak Perfect Bayesian Equilibrium (WPBE) of the game. In contrast, when signals of traders are unconditionally independent, truthful betting

Abstract. We study the strategic behavior of risk-neutral non-myopic agents in Dynamic Parimutuel Markets (DPM). In a DPM, agents buy or sell shares of contracts, whose future payoff in a particular state depends on aggregated trades of all agents. A forward-looking agent hence takes into consideration of possible future trades of other agents when making its trading decision. In this paper, we analyze non-myopic strategies in a two-outcome DPM under a simple model of incomplete information and examine whether an agent will truthfully reveal its information in the market. Specifically, we first characterize a single agent’s optimal trading strategy given the payoff uncertainty. Then, we use a two-player game to examine whether an agent will truthfully reveal its information when it only participates in the market once. We prove that truthful betting is a Nash equilibrium of the two-stage game in our simple setting for uniform initial market probabilities. However, we show that there exists some initial market probabilities at which the first player has incentives to mislead the other agent in the two-stage game. Finally, we briefly discuss when an agent can participate more than once in the market whether it will truthfully reveal its information at its first play in a three-stage game. We find that in some occasions truthful betting is not a Nash equilibrium of the three-stage game even for uniform initial market probabilities. 1