Non-Cooperative Rational Interactive Proofs

Interactive-proof games model the scenario where an honest party interacts with powerful but strategic provers, to elicit from them the correct answer to a computational question. Interactive proofs are increasingly used as a framework to design protocols for computation outsourcing. Existing interactive-proof games largely fall into two categories: either as games of cooperation such as multi-prover interactive proofs and cooperative rational proofs, where the provers work together as a team; or as games of conflict such as refereed games, where the provers directly compete with each other in a zero-sum game. Neither of these extremes truly capture the strategic nature of service providers in outsourcing applications. How to design and analyze non-cooperative interactive proofs is an important open problem. In this paper, we introduce a mechanism-design approach to define a multi-prover interactive-proof model in which the provers are rational and non-cooperative - they act to maximize their expected utility given others\u27 strategies. We define a strong notion of backwards induction as our solution concept to analyze the resulting extensive-form game with imperfect information. We fully characterize the complexity of our proof system under different utility gap guarantees. (At a high level, a utility gap of u means that the protocol is robust against provers that may not care about a utility loss of 1/u.) We show, for example, that the power of non-cooperative rational interactive proofs with a polynomial utility gap is exactly equal to the complexity class P^{NEXP}

Rational Proofs with Multiple Provers

Interactive proofs (IP) model a world where a verifier delegates computation to an untrustworthy prover, verifying the prover's claims before accepting them. IP protocols have applications in areas such as verifiable computation outsourcing, computation delegation, cloud computing. In these applications, the verifier may pay the prover based on the quality of his work. Rational interactive proofs (RIP), introduced by Azar and Micali (2012), are an interactive-proof system with payments, in which the prover is rational rather than untrustworthy---he may lie, but only to increase his payment. Rational proofs leverage the provers' rationality to obtain simple and efficient protocols. Azar and Micali show that RIP=IP(=PSAPCE). They leave the question of whether multiple provers are more powerful than a single prover for rational and classical proofs as an open problem. In this paper, we introduce multi-prover rational interactive proofs (MRIP). Here, a verifier cross-checks the provers' answers with each other and pays them according to the messages exchanged. The provers are cooperative and maximize their total expected payment if and only if the verifier learns the correct answer to the problem. We further refine the model of MRIP to incorporate utility gap, which is the loss in payment suffered by provers who mislead the verifier to the wrong answer. We define the class of MRIP protocols with constant, noticeable and negligible utility gaps. We give tight characterization for all three MRIP classes. We show that under standard complexity-theoretic assumptions, MRIP is more powerful than both RIP and MIP ; and this is true even the utility gap is required to be constant. Furthermore the full power of each MRIP class can be achieved using only two provers and three rounds. (A preliminary version of this paper appeared at ITCS 2016. This is the full version that contains new results.)Comment: Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science. ACM, 201

Complex Economic Systems: Using Collective Intentionality Analysis to Explain Individual ldentity in Networks

Une approche particulière de l\u27analyse de économies vues comme des systèmes complexes étudie l\u27interaction entre les individus dans des réseaus locaux ou de voisinage qui sont un sous-ensemble des économies plus larges. En rejetant la vision traditionnelle des fondations microéconomiques de la relation entre le comportement économique individuel et agrégé comme s\u27influençant réciproquement. Ce papier étudie la conception des réseaux de l\u27individu interactif utilisée dans l\u27analyse d\u27Alan Kirman (2001) dans le cadre de relations de loyauté entre les acheteurs et les vendeurs sur le marché aux poissons de Marseille, en utilisant le cadre du test d\u27identité que j\u27ai appliqué précédemment à la conception atomistique standard de l\u27individu [Davis (2003c)]. Pour ce faire, le papier interprète l\u27individu interactif dans les termes de l\u27analyse de l\u27intention collective et d\u27engagements communs tel qu\u27ils ont été considérés par Margaret Gilbert. Il donne alors en premier lieu une explication sur la formation par les acheteurs et vendeurs d\u27engagements communs, tout restant quand même des individus distincts, et, deuxièmement, argumente qu\u27à travers le temps les individus ainsi compris peuvent également être reidentifiés comme des être distincts. Le papier montre ainsi que le cadre d\u27analyse des réseaux présente une approche pertinente des individus compris en termes de relations sociales qui émergent au travers d\u27engagements communs. One approach to the analysis of economies as complex systems investigates interaction between individuals in local networks or neighborhoods that are subsets of larger economies. Rejecting the traditional microfoundations view of the relation between individual and aggregate economic behavior, network approaches explain individual and aggregate behavior as mutually influencing. This paper investigates the network conception of the interactive individual as employed in Alan Kirman\u27s (2001) analysis of loyalty relationships between buyers and sellers in the Marseille fish market using the identity test framework I previously applied to the standard atomistic conception of the individual [Davis (2003c)]. To do so, the paper interprets the interactive individual in terms of collective intentionality analysis and joint commitments, as understood by Margaret Gilbert. It then, first, gives an explanation of how buyers and sellers can form joint commitments and yet still remain distinct individuals, and, second, argues that over time individuals thus understood can also be re-identified as distinct individuals. The paper thus presents the network framework as offering a viable account of individuals understood in terms of social relationships that emerge out of joint commitments

Common reasoning in games: a Lewisian analysis of common knowledge of rationality

The game-theoretic assumption of ‘common knowledge of rationality’ leads to paradoxes when rationality is represented in a Bayesian framework as cautious expected utility maximisation with independent beliefs (ICEU). We diagnose and resolve these paradoxes by presenting a new class of formal models of players’ reasoning, inspired by David Lewis’s account of common knowledge, in which the analogue of common knowledge is derivability in common reason. We show that such models can consistently incorporate any of a wide range of standards of decision-theoretic practical rationality. We investigate the implications arising when the standard of decision-theoretic rationality so assumed is ICEU.Common reasoning; common knowledge; common knowledge of rationality; David Lewis; Bayesian models of games

Rational proofs

We study a new type of proof system, where an unbounded prover and a polynomial time verifier interact, on inputs a string x and a function f, so that the Verifier may learn f(x). The novelty of our setting is that there no longer are "good" or "malicious" provers, but only rational ones. In essence, the Verifier has a budget c and gives the Prover a reward r ∈ [0,c] determined by the transcript of their interaction; the prover wishes to maximize his expected reward; and his reward is maximized only if he the verifier correctly learns f(x). Rational proof systems are as powerful as their classical counterparts for polynomially many rounds of interaction, but are much more powerful when we only allow a constant number of rounds. Indeed, we prove that if f ∈ #P, then f is computable by a one-round rational Merlin-Arthur game, where, on input x, Merlin's single message actually consists of sending just the value f(x). Further, we prove that CH, the counting hierarchy, coincides with the class of languages computable by a constant-round rational Merlin-Arthur game. Our results rely on a basic and crucial connection between rational proof systems and proper scoring rules, a tool developed to elicit truthful information from experts.United States. Office of Naval Research (Award number N00014-09-1-0597

When All is Said and Done, How Should You Play and What Should You Expect?

Modern game theory was born in 1928, when John von Neumann published his Minimax Theorem. This theorem ascribes to all two-person zero-sum games a value–what rational players may expect–and optimal strategies–how they should play to achieve that expectation. Seventyseven years later, strategic game theory has not gotten beyond that initial point, insofar as the basic questions of value and optimal strategies are concerned. Equilibrium theories do not tell players how to play and what to expect; even when there is a unique Nash equilibrium, it it is not at all clear that the players “should” play this equilibrium, nor that they should expect its payoff. Here, we return to square one: abandon all ideas of equilibrium and simply ask, how should rational players play, and what should they expect. We provide answers to both questions, for all n-person games in strategic form.