Simplified three player Kuhn poker

We study a very small three player poker game (one-third street Kuhn poker), and a simplified version of the game that is interesting because it has three distinct equilibrium solutions. For one-third street Kuhn poker, we are able to find all of the equilibrium solutions analytically. For large enough pot size, $P$, there is a degree of freedom in the solution that allows one player to transfer profit between the other two players without changing their own profit. This has potentially interesting consequences in repeated play of the game. We also show that in a simplified version of the game with $P>5$, there is one equilibrium solution if $5 < P < P^* \equiv (5+\sqrt{73})/2$, and three distinct equilibrium solutions if $P > P^*$. This may be the simplest non-trivial multiplayer poker game with more than one distinct equilibrium solution and provides us with a test case for theories of dynamic strategy adjustment over multiple realisations of the game. We then study a third order system of ordinary differential equations that models the dynamics of three players who try to maximise their expectation by continuously varying their betting frequencies. We find that the dynamics of this system are oscillatory, with two distinct types of solution. We then study a difference equation model, based on repeated play of the game, in which each player continually updates their estimates of the other players' betting frequencies. We find that the dynamics are noisy, but basically oscillatory for short enough estimation periods and slow enough frequency adjustments, but that the dynamics can be very different for other parameter values.Comment: 41 pages, 2 Tables, 17 Figure

Self-Enforcing Climate Change Treaties: A Generalized Differential Game Approach with Applications

Based on recent proposals on non cooperative dynamic games for analysing climate negotiation outcomes, such as Dutta and Radner (2004, 2006a), we generalize a specific framework for modelling differential games of this type and describe the set of conditions for the existence of closed loop dynamics and its relation to adaptive evolutionary dynamics. We then show that the Dutta and Radner (2004, 2006a) discrete time dynamic setup is a specific case of that generalization and describe the dynamics both analytically and numerically for closed loop feedback and perfect state patterns. Our discussion is completed with the introduction of a cooperative differential framework for welfare analysis purposes, within our non cooperative proposal for climate negotiations.Differential Game Theory, Environmental Economics, Evolutionary Dynamics, Climate Change Treaties

Gravity as an emergent phenomenon: a GFT perspective

While the idea of gravity as an emergent phenomenon is an intriguing one, little is known about concrete implementations that could lead to viable phenomenology, most of the obstructions being related to the intrinsic difficulties of formulating genuinely pregeometric theories. In this paper we present a preliminary discussion of the impact of critical behavior of certain microscopic models for gravity, based on group field theories, on the dynamics of the macroscopic regime. The continuum limit is examined in light of some scaling assumption, and the relevant consequences for low energy effective theories are discussed, the role of universality, the corrections to scaling, the emergence of gravitational theories and the nature of their thermodynamical behavior.Comment: 1+26 page