Products of Representations Characterize the Products of Dispersions and the Consistency of Beliefs

A "dispersion" specifies the relative probability between any two elements of a finite domain. It thereby partitions the domain into equivalence classes separated by infinite relative probability. The paper's novelty is to numerically represent not only the order of the equivalence classes, but also the "magnitude" of the gaps between them. The paper's main theorem is that the many products of two dispersions are characterized algebraically by varying the magnitudes of the gaps between each factor's equivalence classes. An immediate corollary is that the many beliefs consistent with two strategies are characterized by varying each player's "steadiness" in avoiding various zero-probability optionsconsistent beliefs, relative probability

The Category of Node-and-Choice Extensive-Form Games

This paper develops the category $\mathbf{NCG}$. Its objects are node-and-choice games, which include essentially all extensive-form games. Its morphisms allow arbitrary transformations of a game's nodes, choices, and players, as well as monotonic transformations of the utility functions of the game's players. Among the morphisms are subgame inclusions. Several characterizations and numerous properties of the isomorphisms are derived. For example, it is shown that isomorphisms preserve the game-theoretic concepts of no-absentmindedness, perfect-information, and (pure-strategy) Nash-equilibrium. Finally, full subcategories are defined for choice-sequence games and choice-set games, and relationships among these two subcategories and $\mathbf{NCG}$ itself are expressed and derived via isomorphic inclusions and equivalences.Comment: 49 pages, 10 figures; revision makes only expositional changes (an improved introduction and a new running example

A pleasant homeomorphism for conditional probability systems

Suppose that there are n states, each denoted by i Є {1,2,...n}. This paper shows that the function [pi|E]i Є E → [Σj ≠ i pi|{i, j}] i is a homeomorphism from the set of conditional probability systems onto the convex hull of all permutations of the n-dimensional vector (0,1,2,...n-1)

Two characterizations of consistency

This paper offers two characterizations of the Kreps-Wilson concept of consistent beliefs. One is primarily of applied interest: beliefs are consistent iff they can be constructed by multiplying together vectors of monomials which induce the strategies. The other is primarily of conceptual interest: beliefs are consistent iff they can be induced by a product dispersion whose marginal dispersions induce the strategies (a dispersion is defined as a relative probability system, and a product dispersion is defined as a joint dispersion whose marginal dispersions are independent). Both these characterizations are derived with linear algebra

A comment on "Sequential Equilibria"

[Introduction] In addition to introducing the path-breaking concept of sequential equilibrium, Kreps and Wilson (1982) (henceforth KW) contains three insightful theorems which derive the geometry of the set of sequential equilibrium assessments, the finiteness of the set of sequential equilibrium outcomes, and the perfection of strict sequential equilibria. These derivations depend upon Lemmas A1 and A2 in the KW appendix. Section 3 of this paper notes that the KW proofs of these lemmas contain a nontrivial fallacy. Section 4 repairs these proofs by means of Streufert (2006b)

Specifying nodes as sets of choices

Osborne and Rubinstein (1994) specify each node in a game tree as a sequence of actions. It is well-known that such actions can be replaced by choices (i.e. agent-specific actions) without loss of generality. I find that this sequential formulation is redundant in the sense that nodes can be equivalently specified as sets of choices. The only cost of doing so is to rule out absent-mindedness. My analysis encompasses both ordered and unordered information sets and both finite and infinite horizons. (This specification of nodes as sets of choices differs from the literature's specification of nodes as sets of outcomes.

Concisely specifying choices in an outcome-set form

Von Neumann and Morgenstern (1944) specify both nodes and choices as sets of outcomes. This outcome-set formulation is extended to the infinite horizon by the discrete extensive forms of Alós-Ferrer and Ritzberger (2013). I propose to restrict such outcome-set forms with a new assumption called "conciseness". Conciseness requires that choices be defined in an economical fashion. I find broad classes of infinite horizon forms that violate conciseness. Yet, I show that every outcome-set form can be equivalently re-defined so as to satisfy conciseness. Thus the assumption of conciseness can increase mathematical tractability at no cost to game theorists

Specifying nodes as sets of actions

The nodes of an extensive-form game are commonly specified as sequences of actions. Rubinstein calls such nodes histories. We find that this sequential notation is superfluous in the sense that nodes can also be specified as sets of actions. The only cost of doing so is to rule out games with absent-minded agents. Our set-theoretic analysis accommodates general finite-horizon games with arbitrarily large action spaces and arbitrarily configured information sets. One application is Streufert (2012), which specifies nodes as sets in order to formulate and prove new results about Kreps-Wilson consistency

The Category of Node-and-Choice Forms, with Subcategories for Choice-Sequence Forms and Choice-Set Forms

The literature specifies extensive-form games in many styles, and eventually I hope to formally translate games across those styles. Toward that end, this paper defines $\mathbf{NCF}$, the category of node-and-choice forms. The category's objects are extensive forms in essentially any style, and the category's isomorphisms are made to accord with the literature's small handful of ad hoc style equivalences. Further, this paper develops two full subcategories: $\mathbf{CsqF}$ for forms whose nodes are choice-sequences, and $\mathbf{CsetF}$ for forms whose nodes are choice-sets. I show that $\mathbf{NCF}$ is "isomorphically enclosed" in $\mathbf{CsqF}$ in the sense that each $\mathbf{NCF}$ form is isomorphic to a $\mathbf{CsqF}$ form. Similarly, I show that $\mathbf{CsqF_{\tilde a}}$ is isomorphically enclosed in $\mathbf{CsetF}$ in the sense that each $\mathbf{CsqF}$ form with no-absentmindedness is isomorphic to a $\mathbf{CsetF}$ form. The converses are found to be almost immediate, and the resulting equivalences unify and simplify two ad hoc style equivalences in Kline and Luckraz 2016 and Streufert 2019. Aside from the larger agenda, this paper already makes three practical contributions. Style equivalences are made easier to derive by [1] a natural concept of isomorphic invariance and [2] the composability of isomorphic enclosures. In addition, [3] some new consequences of equivalence are systematically deduced.Comment: 43 pages, 9 figure

Believing in Multiple Equilibria

If agents have common priors concerning the probability with which equilibrium is selected, they have an incentive to trade beforehand. If their trading process satisfies a certain notion of individual rationality, these trades will reduce and ultimately remove all uncertainty concerning equilibrium selection. In this sense, multiple equilibria engender institutions which can uniquely determine equilibrium