Program Equilibria and Discounted Computation Time

Tennenholtz (GEB 2004) developed Program Equilibrium to model play in a finite two-player game where each player can base their strategy on the other player's strategies. Tennenholtz's model allowed each player to produce a "loop-free" computer program that had access to the code for both players. He showed a folk theorem where any mixed-strategy individually rational play could be an equilibrium payo in this model even in a one-shot game. Kalai et al. gave a general folk theorem for correlated play in a more generic commitment model. We develop a new model of program equilibrium using general computational models and discounting the payos based on the computation time used. We give an even more general folk theorem giving correlated-strategy payoffs down to the pure minimax of each player. We also show equilibrium in other games not covered by the earlier work.brokers, applied mechanism design, linear commission fees, optimal indirect mechanisms, internet auctions, auction houses.

Multi-outcome and Multidimensional Market Scoring Rules

Hanson's market scoring rules allow us to design a prediction market that still gives useful information even if we have an illiquid market with a limited number of budget-constrained agents. Each agent can "move" the current price of a market towards their prediction. While this movement still occurs in multi-outcome or multidimensional markets we show that no market-scoring rule, under reasonable conditions, always moves the price directly towards beliefs of the agents. We present a modified version of a market scoring rule for budget-limited traders, and show that it does have the property that, from any starting position, optimal trade by a budget-limited trader will result in the market being moved towards the trader's true belief. This mechanism also retains several attractive strategic properties of the market scoring rule

Bounding Rationality by Discounting Time

Consider a game where Alice generates an integer and Bob wins if he can factor that integer. Traditional game theory tells us that Bob will always win this game even though in practice Alice will win given our usual assumptions about the hardness of factoring. We define a new notion of bounded rationality, where the payoffs of players are discounted by the computation time they take to produce their actions. We use this notion to give a direct correspondence between the existence of equilibria where Alice has a winning strategy and the hardness of factoring. Namely, under a natural assumption on the discount rates, there is an equilibriumwhere Alice has a winning strategy iff there is a linear-time samplable distribution with respect to which Factoring is hard on average. We also give general results for discounted games over countable action spaces, including showing that any game with bounded and computable payoffs has an equilibrium in our model, even if each player is allowed a countable number of actions. It follows, for example, that the Largest Integer game has an equilibrium in our model though it has no Nash equilibria or epsilon-Nash equilibria.Comment: To appear in Proceedings of The First Symposium on Innovations in Computer Scienc

Bounding Rationality by Discounting Time

Consider a game where Alice generates an integer and Bob wins if he can factor that integer. Traditional game theory tells us that Bob will always win this game even though in practice Alice will win given our usual assumptions about the hardness of factoring. We define a new notion of bounded rationality, where the payoffs of players are discounted by the computation time they take to produce their actions. We use this notion to give a direct correspondence between the existence of equilibria where Alice has a winning strategy and the hardness of factoring. Namely, under a natural assumption on the discount rates, there is an equilibriumwhere Alice has a winning strategy iff there is a linear-time samplable distribution with respect to which Factoring is hard on average. We also give general results for discounted games over countable action spaces, including showing that any game with bounded and computable payoffs has an equilibrium in our model, even if each player is allowed a countable number of actions. It follows, for example, that the Largest Integer game has an equilibrium in our model though it has no Nash equilibria or E-Nash equilibria.Bounded rationality; Discounting; Uniform equilibria; Factoring game

Derandomizing from Random Strings

In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings R_K. It was previously known that PSPACE, and hence BPP is Turing-reducible to R_K. The earlier proof relied on the adaptivity of the Turing-reduction to find a Kolmogorov-random string of polynomial length using the set R_K as oracle. Our new non-adaptive result relies on a new fundamental fact about the set R_K, namely each initial segment of the characteristic sequence of R_K is not compressible by recursive means. As a partial converse to our claim we show that strings of high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings

Two queries

AbstractWe consider the question whether two queries SAT are as powerful as one query. We show that if PNP[1]=PNP[2] then: Locally either NP=coNP or NP has polynomial-size circuits; PNP=PNP[1]; Σp2⊆Πp2/1; Σp2=UPNP[1]∩RPNP[1]; PH=BPPNP[1]. Moreover, we extend the work of Hemaspaandra, Hemaspaandra, and Hempel to show that if PΣp2[1]=PΣp2[2] then Σp2=Πp2. We also give a relativized world, where PNP[1]=PNP[2], but NP≠coNP