arXiv subjects

Peter A. Streufert

Publications and source records attributed to Peter A. Streufert.

5 recordsLinked to original sources

arXiv2026

The Category Gm for Extensive-Form Games

This paper introduces $\mathbf{Gm}$, which is a category for extensive-form games. The category's objects are extensive-form games, each of which is regarded as a set of nodes which has been endowed with edges, information sets, actions, players, and utility functions. Its morphisms are functions, from source nodes to target nodes, that respect this structure. For instance, a game's information-set collection is newly regarded as a topological basis for the game's decision-node set, and then a morphism's continuity serves to respect the source game's information sets. Given these definitions, a game isomorphism is characterized as a bijection whose restriction to decision nodes is a homeomorphism, whose induced player transformation is bijective, and whose induced run transformation is monotonic with respect to the total preorder determined by each player's utility function.

econ.TH↗

arXiv2024

Specifying a Game-Theoretic Extensive Form as an Abstract 5-ary Relation

This paper specifies an extensive form as a 5-ary relation (that is, as a set of quintuples) which satisfies eight abstract axioms. Each quintuple is understood to list a player, a situation (that is, a name for an information set), a decision node, an action, and a successor node. Accordingly, the axioms are understood to specify abstract relationships between players, situations, nodes, and actions. Such an extensive form is called a "pentaform". Finally, a "pentaform game" is defined to be a pentaform together with utility functions. To ground this new specification in the literature, the paper defines the concept of a "traditional game" to represent the literature's many specifications of finite-horizon and infinite-horizon games. The paper's main result is to construct an intuitive bijection between pentaform games and traditional games. Secondary results concern disaggregating pentaforms by subsets, constructing pentaforms by unions, and initial pentaform applications to Selten subgames and perfect-recall (an extensive application to dynamic programming is in Streufert 2023, arXiv:2302.03855).

econ.TH↗

arXiv2023

Dynamic Programming for Pure-Strategy Subgame Perfection in an Arbitrary Game

This paper uses value functions to characterize the pure-strategy subgame-perfect equilibria of an arbitrary, possibly infinite-horizon game. It specifies the game's extensive form as a pentaform (Streufert 2023p, arXiv:2107.10801v4), which is a set of quintuples formalizing the abstract relationships between nodes, actions, players, and situations (situations generalize information sets). Because a pentaform is a set, this paper can explicitly partition the game form into piece forms, each of which starts at a (Selten) subroot and contains all subsequent nodes except those that follow a subsequent subroot. Then the set of subroots becomes the domain of a value function, and the piece-form partition becomes the framework for a value recursion which generalizes the Bellman equation from dynamic programming. The main results connect the value recursion with the subgame-perfect equilibria of the original game, under the assumptions of upper- and lower-convergence. Finally, a corollary characterizes subgame perfection as the absence of an improving one-piece deviation.

econ.TH↗

arXiv2020

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.

econ.TH↗

arXiv2019

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.

econ.TH↗