arXiv subjects
Publications and source records attributed to Robert Samuel Simon.
A game has approximate equilibria if for every $ε>0$ there is an $ε$-equilibrium. We show that there is a stochastic game that lacks approximate equilibria. This game has finitely many players and actions, their payoffs are Borel measurable functions on the pathways of play, and all players have perfect knowledge of the past histories and the present state.
Given a probability space $(X, {\cal B}, m)$, measure preserving transformations $g_1, \dots , g_k$ of $X$, and a colour set $C$, a colouring rule is a way to colour the space with $C$ such that the colours allowed for a point $x$ are determined by that point's location and the colours of the finitely $g_1 (x), \dots , g_k(x)$ with $g_i(x) \not= x$ for all $i$ and almost all $x$. We represent a colouring rule as a correspondence $F$ defined on $X\times C^k$ with values in $C$. A function $f: X\rightarrow C$ satisfies the rule at $x$ if $f(x) \in F( x, f(g_1 x), \dots , f(g_k x))$. A colouring rule is paradoxical if it can be satisfied in some way almost everywhere with respect to $m$, but not in {\bf any} way that is measurable with respect to a finitely additive measure that extends the probability measure $m$ and for which the finitely many transformations $g_1, \dots , g_k$ remain measure preserving. We show that a colouring rule can be paradoxical when the $g_1, \dots, g_k$ are members of a group $G$, the probability space $X$ and the colour set $C$ are compact sets, $C$ is convex and finite dimensional, and the colouring rule says if $c: X\rightarrow C$ is the colouring function then the colour $c(x)$ must lie ($m$ a.e.) in $F(x, c(g_1(x) ), \dots , c(g_k(x)))$ for a non-empty upper-semi-continuous convex-valued correspondence $F$ defined on $X\times C^k$. We show that any colouring that approximates the correspondence by $ε$ for small enough positive $ε$ cannot be measurable in the same finitely additive way. Furthermore any function satisfying the colouring rule illustrates a paradox through finitely many measure preserving shifts defining injective maps from the whole space to subsets of measure summing up to less than one.
We define "paradoxical colouring rule", show its relation to measure theoretic paradoxes, and demonstrate that proper vertex colouring can be a paradoxical colouring rule.
We colour every point x of a probability space X according to the colours of a finite list x_1, ...., x_k of points such that each of the x_i, as a function of x, is a measure preserving transformation. We ask two questions about a colouring rule (1) does there exist a finitely additive extension of the probability measure for which the x_i remain measure preserving and also a colouring obeying the rule almost everywhere that is measurable with respect to this extension?, and (2) does there exist any colouring obeying the rule almost everywhere? if the answer to the first question is no and to the second question yes, we say that the colouring rule is paradoxical. A paradoxical colouring rule not only allows for a paradoxical partition of the space, it requires one. We pay special attention to generalizations of the Hausdorff paradox.
We present a three player Bayesian game for which there is no epsilon equilibria in Borel measurable strategies for small enough epsilon, however there are non-measurable equilibria.
Let C be a closed subset of a topological space X, and let f : C --> X. Let us assume that f is continuous and f(x) lies in C for every x in the boundary of C. How many times can one iterate f? This paper provides estimates on the number of iterations and examples of their optimality. In particular we show how some topological properties of f, C, X are related to the maximal number of iterations, both in the case of functions and in the more general case of set-valued maps.
We give an alternative proof that every two-person non-zero-sum absorbing positive recursive stochastic game with finitely many states has approximate equilibria, a result proven by Nicolas Vieille. Our proof uses a state specific discount factor which is similar to the conventional discount factor only when there is only one non-absorbing state.
For every set of primitive propositions and agents there is a canonical Kripke structure and a canonical map from any Kripke structure (defined with the same primitive propositions and agents) to this canonical one. A cell of the canonical Kripke structure is a set C such that if any agent considers a point x in C to be possible then all the other points considered possible by this agent are also in C. A cell C has finite fanout if at every point in C every agent considers possible only finitely many other points. We demonstrate a cell of this canonical Kripke structure such that every Kripke structure that maps to this cell does so surjectively, yet this cell does not have finite fanout.
In a game of incomplete information, an infinite state space can create problems. When the space is uncountably large, the strategy spaces of the players may be unwieldly, resulting in a lack of measurable equilibria. When the knowledge of a player allows for an infinite number of possibilities, without conditions on the behavior of the other players, that player may be unable to evaluate and compare the payoff consequences of her actions. We argue that local finiteness is an important and desirable property, namely that at every point in the state space every player knows that only a finite number of points are possible. Local finiteness implies a kind of common knowledge of a countable number of points. Unfortunately its relationship to other forms of common knowledge is complex. In the context of the multi-agent propositional calculus, if the set of formulas held in common knowledge is generated by a finite set of formulas but a finite structure is not determined then there are uncountably many locally finite structures sharing this same set of formulas in common knowledge and likewise uncountably many with uncountable size. This differs radically from the infinite generation of formulas in common knowledge, and we show some examples of this. One corollary is that if there are infinitely many distinct points but a uniform bound on the number of points any player knows is possible then the set of formulas in common knowledge cannot be finitely generated.