2,479 research outputs found
Rationality and Escalation in Infinite Extensive Games
The aim of this of this paper is to study infinite games and to prove
formally some properties in this framework. As a consequence we show that the
behavior (the madness) of people which leads to speculative crashes or
escalation can be fully rational. Indeed it proceeds from the statement that
resources are infinite. The reasoning is based on the concept of coinduction
conceived by computer scientists to model infinite computations and used by
economic agents unknowingly. When used consciously, this concept is not as
simple as induction and we could paraphrase Newton: "Modeling the madness of
people is more difficult than modeling the motion of planets".Comment: arXiv admin note: substantial text overlap with arXiv:1004.5257,
arXiv:0904.3528, and arXiv:0912.174
Intelligent escalation and the principle of relativity
Escalation is the fact that in a game (for instance in an auction), the
agents play forever. The -game is an extremely simple infinite game with
intelligent agents in which escalation arises. It shows at the light of
research on cognitive psychology the difference between intelligence
(algorithmic mind) and rationality (algorithmic and reflective mind) in
decision processes. It also shows that depending on the point of view (inside
or outside) the rationality of the agent may change which is proposed to be
called the principle of relativity.Comment: arXiv admin note: substantial text overlap with arXiv:1306.228
The risk of divergence
We present infinite extensive strategy profiles with perfect information and
we show that replacing finite by infinite changes the notions and the reasoning
tools. The presentation uses a formalism recently developed by logicians and
computer science theoreticians, called coinduction. This builds a bridge
between economic game theory and the most recent advance in theoretical
computer science and logic. The key result is that rational agents may have
strategy leading to divergence .Comment: 3rd International Workshop on Strategic Reasoning, Dec 2015, Oxford,
United Kingdom. 201
On the Rationality of Escalation
Escalation is a typical feature of infinite games. Therefore tools conceived
for studying infinite mathematical structures, namely those deriving from
coinduction are essential. Here we use coinduction, or backward coinduction (to
show its connection with the same concept for finite games) to study carefully
and formally the infinite games especially those called dollar auctions, which
are considered as the paradigm of escalation. Unlike what is commonly admitted,
we show that, provided one assumes that the other agent will always stop,
bidding is rational, because it results in a subgame perfect equilibrium. We
show that this is not the only rational strategy profile (the only subgame
perfect equilibrium). Indeed if an agent stops and will stop at every step, we
claim that he is rational as well, if one admits that his opponent will never
stop, because this corresponds to a subgame perfect equilibrium. Amazingly, in
the infinite dollar auction game, the behavior in which both agents stop at
each step is not a Nash equilibrium, hence is not a subgame perfect
equilibrium, hence is not rational.Comment: 19 p. This paper is a duplicate of arXiv:1004.525
Coalgebraic analysis of subgame-perfect equilibria in infinite games without discounting
We present a novel coalgebraic formulation of infinite extensive games. We define both the game trees and the strategy profiles by possibly infinite systems of corecursive equations.
Certain strategy profiles are proved to be subgame perfect equilibria using a novel proof principle of predicate coinduction which is shown to be sound by reducing it to Kozen’s metric coinduction. We characterize all subgame perfect equilibria for the dollar auction game. The economically interesting feature is that in order to prove these results we do not need to rely on continuity assumptions on the payoffs which amount to discounting the future.
In particular, we prove a form of one-deviation principle without any such assumptions. This suggests that coalgebra supports a more adequate treatment of infinite-horizon models in game theory and economics
Deconstruction of Infinite Extensive Games using coinduction
Finite objects and more specifically finite games are formalized using
induction, whereas infinite objects are formalized using coinduction. In this
article, after an introduction to the concept of coinduction, we revisit on
infinite (discrete) extensive games the basic notions of game theory. Among
others, we introduce a definition of Nash equilibrium and a notion of subgame
perfect equilibrium for infinite games. We use those concepts to analyze well
known infinite games, like the dollar auction game and the centipede game and
we show that human behaviors that are often considered as illogic are perfectly
rational, if one admits that human agents reason coinductively.Comment: 19
Infinite subgame perfect equilibrium in the Hausdorff difference hierarchy
Subgame perfect equilibria are specific Nash equilibria in perfect
information games in extensive form. They are important because they relate to
the rationality of the players. They always exist in infinite games with
continuous real-valued payoffs, but may fail to exist even in simple games with
slightly discontinuous payoffs. This article considers only games whose outcome
functions are measurable in the Hausdorff difference hierarchy of the open sets
(\textit{i.e.} when in the Baire space), and it characterizes the
families of linear preferences such that every game using these preferences has
a subgame perfect equilibrium: the preferences without infinite ascending
chains (of course), and such that for all players and and outcomes
we have . Moreover at
each node of the game, the equilibrium constructed for the proof is
Pareto-optimal among all the outcomes occurring in the subgame. Additional
results for non-linear preferences are presented.Comment: The alternative definition of the difference hierarchy has changed
slightl
Game theory of mind
This paper introduces a model of ‘theory of mind’, namely, how we represent the intentions and goals of others to optimise our mutual interactions. We draw on ideas from optimum control and game theory to provide a ‘game theory of mind’. First, we consider the representations of goals in terms of value functions that are prescribed by utility or rewards. Critically, the joint value functions and ensuing behaviour are optimised recursively, under the assumption that I represent your value function, your representation of mine, your representation of my representation of yours, and so on ad infinitum. However, if we assume that the degree of recursion is bounded, then players need to estimate the opponent's degree of recursion (i.e., sophistication) to respond optimally. This induces a problem of inferring the opponent's sophistication, given behavioural exchanges. We show it is possible to deduce whether players make inferences about each other and quantify their sophistication on the basis of choices in sequential games. This rests on comparing generative models of choices with, and without, inference. Model comparison is demonstrated using simulated and real data from a ‘stag-hunt’. Finally, we note that exactly the same sophisticated behaviour can be achieved by optimising the utility function itself (through prosocial utility), producing unsophisticated but apparently altruistic agents. This may be relevant ethologically in hierarchal game theory and coevolution
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