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Abstract:
This paper describes ways that the definition of an equilibrium among players' strategies in a game can be sharpened by invoking additional criteria derived from decision theory. Refinements of John Nash's 1950 definition aim primarily to distinguish equilibria in which implicit commitments are credible due to incentives. One group of refinements requires sequential rationality as the game progresses. Another ensures credibility by considering perturbed games in which every contingency occurs with positive probability, which has the further advantage of excluding weakly dominated strategies.

economic theory

Abstract:
A player's pure strategy is called relevant for an outcome of a game in extensive form with perfect recall if there exists a weakly sequential equilibrium with that outcome for which the strategy is an optimal reply at every information set it does not exclude. The outcome satisfies forward induction if it results from a weakly sequential equilibrium in which players' beliefs assign positive probability only to relevant strategies at each information set reached by a profile of relevant strategies. We prove that if there are two players and payoffs are generic then an outcome satisfies forward induction if every game with the same reduced normal form after eliminating redundant pure strategies has a sequential equilibrium with an equivalent outcome. Thus in this case forward induction is implied by decision-theoretic criteria.

Abstract:
This article characterizes a supply function equilibrium in an auction market constrained by limited capacities of links in a transportation network and limited input/output capacities of participants. The formulation is adapted to a wholesale spot market for electricity managed by the operator of the transmission system. The results are derived using the calculus of variations to obtain the Euler conditions and the transversality conditions that characterize a Nash equilibrium in an auction in which bids are as supply functions, and quantities and payments are based either on nodal prices or pay-as-bid.

Abstract:
The Global Newton Method for games in normal form and in extensive form is shown to have a natural extension to computing Markov-perfect equilibria of stochastic games.

Abstract:
We examine Hillas and Kohlberg's conjecture that invariance to the addition of payoff-redundant strategies implies that a backward induction outcome survives deletion of strategies that are inferior replies to all equilibria with the same outcome. That is, invariance and backward induction imply forward induction. Although it suffices in simple games to interpret backward induction as a subgame-perfect or sequential equilibrium, to obtain general theorems we use a quasi-perfect equilibrium, viz. a sequential equilibrium in strategies that are admissible continuations from each information set. Using this version of backward induction, we prove the Hillas-Kohlberg conjecture for two-player extensive-form games with perfect recall. We also prove an analogous theorem for general games by interpreting backward induction as a proper equilibrium, since a proper equilibrium is equivalent to a quasi-perfect equilibrium of each extensive form with the same normal form, provided beliefs are justifed by perturbations invariant to inessential transformations of the extensive form. For a two-player game we prove that if a set of equilibria includes a proper equilibrium of every game with the same reduced normal form then it satisfies forward induction, i.e. it includes a proper equilibrium of the game after deleting strategies that are inferior replies to all equilibria in the set. We invoke slightly stronger versions of invariance and properness to handle nonlinearities in an N-player game.

economic theory, game theory

Abstract:
An N-player game can be approximated by adding a coordinator who interacts bilaterally with each player. The coordinator proposes strategies to the players, and his payoff is maximized when each player's optimal reply agrees with his proposal. When the feasible set of proposals is finite, a solution of an associated linear complementarity problem yields an approximate equilibrium of the original game. Computational efficiency is improved by using the vertices of Kuhn's triangulation of the players' strategy space for the coordinator's pure strategies. Computational experience is reported.

economic theory, game theory

Abstract:
We define a refinement of Nash equilibria called metastability. This refinement supposes that the given game might be embedded within any global game that leaves its local bestreply correspondence unaffected. A selected set of equilibria is metastable if it is robust against perturbations of every such global game; viz., every sufficiently small perturbation of the best-reply correspondence of each global game has an equilibrium that projects arbitrarily near the selected set. Metastability satisfies the standard decision-theoretic axioms obtained by Mertens' (1989) refinement (the strongest proposed refinement), and it satisfies the projection property in Mertens' small-worlds axiom: a metastable set of a global game projects to a metastable set of a local game. But the converse is slightly weaker than Mertens' decomposition property: a metastable set of a local game contains a metastable set that is the projection of a metastable set of a global game. This is inevitable given our demonstration that metastability is equivalent to a strong form of homotopic essentiality. Mertens' definition invokes homological essentiality whereas we derive homotopic essentiality from primitives (robustness for every embedding). We argue that this weak version of decomposition has a natural gametheoretic interpretation.

economic theory, game theory

Abstract:
Two assumptions are used to justify selection of equilibria in stable sets. One assumption requires that a selected set is invariant to addition of redundant strategies. The other is a strong version of backward induction. Backward induction is interpreted as the requirement that behavior strategies in an extensive-form game are sequentially rational and conditionally admissible at every information set; viz., quasi-perfect as defined by van Damme. The strong version requires 'truly' quasi-perfect, in that every action perturbation selects a quasi-perfect equilibrium in the set. For two-player games we also provide an exact characterization of stable sets.

Economic theory, game theory

Abstract:
The connected uniformly-hyperstable sets of a finite game are precisely the essential components of Nash equilibria.

economic theory

Abstract:
From a variant of Kuhn's triangulation we derive a discrete version of the Global Newton Method that yields an epsilon-equilibrium of an N-player game and then sequentially reduces epsilon toward zero to obtain any desired precision or the best precision for any number of iterations.

Abstract:
Mertens' (1989) definition of stability for a game in strategic form is applied to a game in extensive form with perfect recall. If payoffs are generic then the outcomes of stable sets of equilibria defined via homological essentiality by Mertens coincide with those defined via homotopic essentiality. This implies that for such games various definitions of stability in terms of perturbations of players' strategies as in Mertens, or best-reply correspondences as in Hillas (1990), yield the same outcomes. In proving this result it is convenient, as a practical matter, to work with enabling strategies, which are like behavioral strategies, instead of mixed strategies. A corollary yields a computational test that usually suffices to identify the stable outcomes of such a game.

economic theory, game theory, extensive form, generic, stableset, computation

Abstract:
Three axioms from decision theory select sets of Nash equilibria of signaling games in extensive form with generic payoffs. The axioms require undominated strategies (admissibility), inclusion of a sequential equilibrium (backward induction), and dependence only on the game's normal form even when embedded in a larger game with redundant strategies or irrelevant players(small worlds). The axioms are satisfied by a set that is stable (Mertens, 1989) and conversely the axioms imply that each selected set is stable and thus an essential component of admissible equilibria with the same outcome.

Abstract:
We apply three axioms adapted from decision theory to refinements of the Nash equilibria of games with perfect recall that select connected closed subsets called solutions. Undominated Strategies: No player uses a weakly dominated strategy in an equilibrium in a solution. Backward Induction: Each solution contains a quasi-perfect equilibrium and thus a sequential equilibrium in strategies that provide conditionally admissible optimal continuations from information sets. Small Worlds: A refinement is immune to embedding a game in a larger game with additional players provided the original players' strategies and payoffs are preserved, i.e. solutions of a game are the same as those induced by the solutions of any larger game in which it is embedded. For games with two players and generic payoffs, we prove that these axioms characterize each solution as an essential component of equilibria in undominated strategies, and thus a stable set as defined by Mertens (1989).

Abstract:
Three axioms from decision theory are applied to refinements that select connected subsets of the Nash equilibria of games with perfect recall. The first axiom requires all equilibria in a selected subset to be admissible, i.e. each player's strategy is an admissible optimal reply to other players' strategies. The second axiom invokes backward induction by requiring a selected subset to contain a sequential equilibrium. The third axiom requires a refinement to be immune to embedding a game in a larger game with additional strategies and players, provided the original players' strategies and payoffs are preserved, viz., selected subsets must be the same as those induced by the selected subsets of any larger game in which it is embedded. These axioms are satisfied by refinements that select subsets that are stable as defined by Mertens (1989).

For a game with two players, perfect information, and generic payoffs, we prove the converse that the axioms require a selected set to be stable. In the space of mixed strategies of minimal dimension, the stable set is unique and consists of the admissible equilibria with the same outcome as the unique subgame-perfect equilibrium obtained by backward induction. Each other admissible equilibrium with this outcome is the profile of players' strategies in an admissible sequential equilibrium of a larger game in which the original game is embedded, so the third axiom requires it to be included.

economic theory

Abstract:
The sequential equilibrium of an ascending-price auction of a single item is derived explicitly for the case of log-normal distributions and a multiplicative valuation model comprising both common and private factors, and allowing asymmetries. If the prior distribution on the common factors is diffuse, or of the form obtained by Bayesian updating from a diffuse prior distribution, then the equilibrium strategies are log-linear with coefficients obtained by solving a set of linear equations. A similar construction applies to normal distributions and additive terms in the valuation model. An example illustrates the predictions derived from the model.

Abstract:
We construct a model of pre-trial discovery and settlement negotiations in which, in reverse order: the trial is a zero-sum game that each party incurs a cost to play; the parties can avoid trial and its costs by settling on a payment from the defendant to the plaintiff; and before negotiating settlement, each party can discover imperfect sample information about the facts known privately by the other, where such discovery imposes costs on both parties. We use a linear model of the trial-game value as a function of the parties' one-dimensional privately known parameters; we use a mechanism-design formulation to ensure that the settlement process maximizes the expected avoided cost of trial; we use a Normal diffusion process (Brownian motion) to model the acquisition of sample information via discovery; and we assume that the parties' privately known parameters are drawn independently from a diffuse Normal distribution. The main result shows that the parties' ex ante expected gains from a joint plan of discovery are unaffected by their privately known parameters; hence, from a Coasian viewpoint, there is no intrinsic impediment to agreement initially on an efficient plan of discovery. That is, even though there are informational disparities between the parties, these have no effect on each party's calculation of the expected benefits and costs of a discovery plan, and therefore need not impede agreement on an efficient plan. However, a dynamic process of discovery is problematic because the parties' incentives at later stages are affected by the outcomes of prior discoveries, and indeed the amounts of discoveries can be strategic substitutes or complements depending on the prior evolution of discoveries. This indicates the importance of agreement in the initial conference with the judge on an efficient plan of discovery, with subsequent enforcement based on contingent strategies in which excessive discovery by one party precipitates further discovery by the other. These results for this particular model indicate that discovery is not inherently inefficient, but they also indicate that 'wars of discovery' are possible, especially when discoveries are undertaken sequentially. A subsidiary result is that each discovery benefits both parties (gross of the costs imposed) by reducing the risk of trial, although the discovering party tends to get the greater benefit because the discovery tilts the terms of settlement in his favor.