Workshop on the Probabilistic Impulse Behind Modern Economic Theory
(11 - 18 Jan 2011)

~ Abstracts ~


A structure theorem for rationalizability in the normal form of dynamic games
Yi-Chun Chen, National University of Singapore

We prove the structure theorem for (interim correlated) rationalizability due to Weinstein and Yildiz (2007) in the normal form of any finite extensive-form game with perfect recall and suitably rich payoffs. Consequently, in these games no refinements of rationalizability are robust in the sense proposed by Weinstein and Yildiz. To obtain this result, we first observe that for the structure theorem it is both necessary and sufficient that every rationalizable action is the unique rationalizable action for some Harsanyi type, and then demonstrate that the condition is satisfied whenever the possible payoffs are sufficiently rich.

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Equilibrium for discontinuous games and optimal regulation in electricity markets
Alejandro Jofré, University of Chile, Chile

In this presentation electricity markets are considered including a transmission network, producers generating electricity and an agent doing coordination (Independent System Operator, ISO). Production is organized by means of an auction. Once producers simultaneously bid cost functions, the ISO decides the quantity each generator produces and the flows through the network lines. Producers play strategically with the ISO. When bidding, each firm tries to obtain revenues as high as possible. We prove existence of equilibrium for this discontinuous game and then by using optimal mechanism design, we derive an optimal regulation mechanism for pricing and compare its performance with the bayesian version of the usual price equal to Lagrange multiplier. Finally, we develop the sensitivity analysis with respect to probabilities distributions involved.

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Belief operators in infinite models
Xiao Luo, National University of Singapore

We follow Morris (JET, 1996) to study the decision-theoretic notion of belief and its logical properties in infinite models; in particular, we extend Morris's main results to the belief operators derived from regular preferences due to Epstein and Wang (Econometrica, 1996). As an application, we formulate and show a general impossibility result of speculative trade in infinite state spaces. Our approach applies to the "probabilistic" notion of belief.

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Equity and incentives in large asymmetric information economies
Stuart McDonald, University of Queensland, Australia

This paper examines the possibility of obtaining an equitable reallocation of resources in a large exchange economy under asymmetric information. In this large economy, if agents? signals are pair-wise independent and negligible in influence, then any incentive compatible allocation is envy-free and can be implemented in a price mechanism with non-zero transfers. Theses allocations will be efficient only if the price mechanism has zero transfers. Implying that perfect competition will exist only in the absence of external economies and diseconomies generated by information asymmetries.

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Recursive multiple temptations
Bin Miao, National University of Singapore

This paper studies a Gul and Pesendorfer (2001; 2004) style model in which decision makers are subject to multiple temptations in every period. I partially show the existence of recursive multiple temptations utility function under some regularity conditions, and the model can account for more preference reversal behaviours that cannot be supported by Gul and Pesendorfer.

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Risk and utility in financial optimization and equilibrium
Terry Rockafellar, University of Washington, USA

Maximizing expected utility is a well appreciated prescription for agents in economics, and there is a tradition of it in mathematical finance as well. In practice, though, most formulations now of financial optimization forgo utility and deal with uncertainty from some other angle.

Quantifications of risk and deviation have a key role in that. Deviation, generalizing standard deviation, refers to some kind of assessment of the nonconstancy of a random variable that represents a future loss (or in a different orientation, gain). Risk in this context refers instead to a value that can be taken as an aggregate for the uncertain future loss, with classical parallels like expected value, worst value, or a particular quantile. Risk and deviation are paired in a tight relationship and can be articulated very satisfyingly in a framework of convex analysis and duality.

Moreover, in this framework, risk can typically be derived by a scheme that determines a best trade-off in preferences between present losses and future losses through consideration of regret, yet another quantification. Regret is a mirror image of utility (in switching from gain to loss by a change of signs). However, it turns out that the orientation for finance should then not be absolute loss but relative loss in comparison with some threshold or benchmark.

Although the impression is widespread in finance that market equilibrium is a consequence of agents all acting only on the basis of expected outcomes and standard deviation, the theoretical support for that is thin. In fact, different agents, perhaps in different legal and fiscal circumstances, could, for example, have preferences in which standard deviation is replaced by some generalized deviation, and the existence of equilibrium can then be established nonetheless.

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Stochastic evolutionary game theory: overview and recent results
William H. Sandholm, University of Wisconsin, USA

We provide an overview of the methods of evolutionary game theory and describe a variety of recent results. Evolutionary game theory provides dynamic models of behavior for populations of agents engaged in recurring strategic interactions. Population games provide a general model of strategic interactions among large numbers of agents; network congestion, multilateral externalities, and natural selection are among their many applications. As the direct assumption of equilibrium play seems difficult to justify in these games, behavior is most naturally modeled as a dynamic adjustment processes. To accomplish this, one begins with an explicit stochastic description of how individual agents make decisions. When the number of agents is large enough and the time horizon of interest not too long, the evolution of aggregate behavior is well approximated by solutions to ordinary differential equations. If one is interested in behavior over very long time spans, one studies the stochastic evolutionary processes directly, focusing on their ergodic and large deviations properties; this is the context for analyses of stochastic stability.

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Independent random partial matching with infinite types
Xiang Sun, National University of Singapore

In this presentation, the mathematical foundation for independent random partial matching of a large population with infinite types, which is widely used in the economics, will be discussed.

There are two parts in the presentation. In part 1, I will review the results for independent random matching with finite types; in part 2, I will introduce the models for independent random matching with infinite types for both static case and dynamic case, and give related results about the existence and exact law of large numbers for them.

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Saturation and games with private information
Yongchao Zhang, National University of Singapore

It is well-known that there is no pure-strategy Nash equilibrium in a finite player game with private information and uncountably infinite actions, if the diffused private information is modeled by a Lebesgue interval; see Khan-Rath-Sun (JME, 1999). In this paper, we provide a thorough answer to this puzzle by working instead with the saturated probability space introduced by Hoover-Keisler (TAMS, 1984). First, the saturation property on private information signals for each player is a sufficient condition for the existence of pure strategy Nash equilibria. Second, given an information structure, if each player's action space is modeled by a uncountable compact metric space, then this saturation property on private information signals also furnishes a necessary condition for the existence of pure strategy Nash equilibria for such games.

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Equilibrium strategies independent of volatility and the size of rivals
Zhixiang Zhang, Central University of Finance and Economics, China

We consider a class of stochastic differential game about renewable common-property resource. It turns out that the volatility and the size of competitors do not affect the players' strategies in our model.

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