Person:

Chen, Yiling

We explore the striking mathematical connections that exist between market scoring rules, cost function based prediction markets, and no-regret learning. We first show that any cost function based prediction market can be interpreted as an algorithm for the commonly studied problem of learning from expert advice by equating the set of outcomes on which bets are placed in the market with the set of experts in the learning setting, and equating trades made in the market with losses observed by the learning algorithm. If the loss of the market organizer is bounded, this bound can be used to derive an (O( \sqrt T)) regret bound for the corresponding learning algorithm. We then show that the class of markets with convex cost functions exactly corresponds to the class of Follow the Regularized Leader learning algorithms, with the choice of a cost function in the market corresponding to the choice of a regularizer in the learning problem. Finally, we show an equivalence between market scoring rules and prediction markets with convex cost functions. This implies both that any market scoring rule can be implemented as a cost function based market maker, and that market scoring rules can be interpreted naturally as Follow the Regularized Leader algorithms. These connections provide new insight into how it is that commonly studied markets, such as the Logarithmic Market Scoring Rule, can aggregate opinions into accurate estimates of the likelihood of future events.

We propose a general framework for the design of securities markets over combinatorial or infinite state or outcome spaces. The framework enables the design of computationally efficient markets tailored to an arbitrary, yet relatively small, space of securities with bounded payoff. We prove that any market satisfying a set of intuitive conditions must price securities via a convex cost function, which is constructed via conjugate duality. Rather than deal with an exponentially large or infinite outcome space directly, our framework only requires optimization over a convex hull. By reducing the problem of automated market making to convex optimization, where many efficient algorithms exist, we arrive at a range of new polynomial-time pricing mechanisms for various problems. We demonstrate the advantages of this framework with the design of some particular markets. We also show that by relaxing the convex hull we can gain computational tractability without compromising the market institution’s bounded budget.

We consider a setting in which a principal seeks to induce an adaptive agent to select a target action by providing incentives on one or more actions. The agent maintains a belief about the value for each action—which may update based on experience—and selects at each time step the action with the maximal sum of value and associated incentive. The principal observes the agent’s selection, but has no information about the agent’s current beliefs or belief update process. For inducing the target action as soon as possible, or as often as possible over a fixed time period, it is optimal for a principal with a per-period budget to assign the budget to the target action and wait for the agent to want to make that choice. But with an across-period budget, no algorithm can provide good performance on all instances without knowledge of the agent’s update process, except in the particular case in which the goal is to induce the agent to select the target action once. We demonstrate ways to overcome this strong negative result with knowledge about the agent’s beliefs, by providing a tractable algorithm for solving the offline problem when the principal has perfect knowledge, and an analytical solution for an instance of the problem in which partial knowledge is available.