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Eilon Solan

Publications and source records attributed to Eilon Solan.

At least 37 records · Page 2Linked to original sources
arXiv2022

Browder's Theorem: from One-Dimensional Parameter Space to General Parameter Space

A parametric version of Brouwer's Fixed Point Theorem, which is proven using the fixed-point index, states that for every continuous mapping $f : (X \times Y) \to Y$, where $X$ is nonempty, compact, and connected subset of a Hausdorff topological space and $Y$ is a nonempty, convex, and compact subset of a locally-convex topological vector space, the set of fixed points of $f$, defined by $C_f := \{ (x,y) \in X \times Y \colon f(x,y)=y\}$, has a connected component whose projection onto the first coordinate is $X$. In this note we provide an elementary proof for this result, using its reduction to the case $X = [0,1]$.

math.GN↗
arXiv2022

Markovian Persuasion with Two States

This paper addresses the question of how to best communicate information over time in order to influence an agent's belief and induced actions in a model with a binary state of the world that evolves according to a Markov process, and with a finite number of actions. We characterize the sender's optimal message strategy in the limit, as the length of each period decreases to zero. The optimal strategy is not myopic. Depending on the agent's beliefs, sometimes no information is revealed, and sometimes the agent's belief is split into two well-chosen posterior beliefs.

econ.TH↗
arXiv2022

Stochastic Games with General Payoff Functions

We consider multiplayer stochastic games in which the payoff of each player is a bounded and Borel-measurable function of the infinite play. By using a generalization of the technique of Martin (1998) and Maitra and Sudderth (1998), we show four different existence results. In each stochastic game, it holds for every $ε>0$ that (i) each player has a strategy that guarantees in each subgame that this player's payoff is at least her maxmin value up to $ε$, (ii) there exists a strategy profile under which in each subgame each player's payoff is at least her minmax value up to $ε$, (iii) the game admits an extensive-form correlated $ε$-equilibrium, and (iv) there exists a subgame that admits an $ε$-equilibrium.

math.OC↗
arXiv2022

Absorbing Blackwell Games

It was shown in Flesch and Solan (2022) with a rather involved proof that all two-player stochastic games with finite state and action spaces and shift-invariant payoffs admit an $ε$-equilibrium, for every $ε>0$. Their proof also holds for two-player absorbing games with tail-measurable payoffs. In this paper we provide a simpler proof for the existence of $ε$-equilibrium in two-player absorbing games with tail-measurable payoffs, by combining recent mathematical tools for such payoff functions with classical tools for absorbing games.

math.OC↗
arXiv2022

Repeated Games with Tail-Measurable Payoffs

We study multiplayer Blackwell games, which are repeated games where the payoff of each player is a bounded and Borel-measurable function of the infinite stream of actions played by the players during the game. These games are an extension of the two-player perfect-information games studied by David Gale and Frank Stewart (1953). Recently, various new ideas have been discovered to study Blackwell games. In this paper, we give an overview of these ideas by proving, in four different ways, that Blackwell games with a finite number of players, finite action sets, and tail-measurable payoffs admit an $\varepsilon$-equilibrium, for all $\varepsilon>0$.

math.OC↗
arXiv2022

Regularity of the minmax value and equilibria in multiplayer Blackwell games

A real-valued function $φ$ that is defined over all Borel sets of a topological space is \emph{regular} if for every Borel set $W$, $φ(W)$ is the supremum of $φ(C)$, over all closed sets $C$ that are contained in $W$, and the infimum of $φ(O)$, over all open sets $O$ that contain $W$. We study Blackwell games with finitely many players. We show that when each player has a countable set of actions and the objective of a certain player is represented by a Borel winning set, that player's minmax value is regular. We then use the regularity of the minmax value to establish the existence of $\varepsilon$-equilibria in two distinct classes of Blackwell games. One is the class of $n$-player Blackwell games where each player has a finite action space and an analytic winning set, and the sum of the minmax values over the players exceeds $n-1$. The other class is that of Blackwell games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values. For the latter class, we obtain a characterization of the set of equilibrium payoffs.

math.OC↗
arXiv2021

Browder's Theorem through Brouwer's Fixed Point Theorem

One of the conclusions of Browder (1960) is a parametric version of Brouwer's Fixed Point Theorem, stating that for every continuous function $f : ([0,1] \times X) \to X$, where $X$ is a simplex in a Euclidean space, the set of fixed points of $f$, namely, the set $\{(t,x) \in [0,1] \times X \colon f(t,x) = x\}$, has a connected component whose projection on the first coordinate is $[0,1]$. Browder's (1960) proof relies on the theory of the fixed point index. We provide an alternative proof to Browder's result using Brouwer's Fixed Point Theorem.

math.GN↗
arXiv2021

Browder's Theorem with General Parameter Space

Browder (1960) proved that for every continuous function $F : X \times Y \to Y$, where $X$ is the unit interval and $Y$ is a nonempty, convex, and compact subset of $\dR^n$, the set of fixed points of $F$, defined by $C_F := \{ (x,y) \in X \times Y \colon F(x,y)=y\}$ has a connected component whose projection to the first coordinate is $X$. We extend this result to the case where $X$ is a connected and compact Hausdorff space.

math.GN↗
arXiv2020

Absorption Paths and Equilibria in Quitting Games

We study quitting games and define the concept of absorption paths, which is an alternative definition to strategy profiles that accomodates both discrete time aspects and continuous time aspects, and is parameterized by the total probability of absorption in past play rather than by time. We then define the concept of sequentially 0perfect absorption paths, which are shown to be limits of $ε$-equilibrium strategy profiles as $ε$ goes to 0. We finally identify a class of quitting games that possess sequentially 0-perfect absorption paths.

math.OC↗
arXiv2020

Sunspot Equilibrium in Positive Recursive Two-Dimensions Quitting Absorbing Games

A uniform sunspot epsilon-equilibrium of a dynamic game is a uniform epsilon-equilibrium in an extended game, where the players observe a public signal at every stage. We prove that a uniform sunspot epsilon-equilibrium exists in two classes of multiplayer absorbing games, thereby extending earlier works by Solan and Solan (2019, 2018).

math.OC↗
arXiv2019

Reachability and safety objectives in Markov decision processes on long but finite horizons

We consider discrete-time Markov decision processes in which the decision maker is interested in long but finite horizons. First we consider reachability objective: the decision maker's goal is to reach a specific target state with the highest possible probability. Formally, strategy $σ$ overtakes another strategy $σ'$, if the probability of reaching the target state within horizon $t$ is larger under $σ$ than under $σ'$, for all sufficiently large $t\in\NN$. We prove that there exists a pure stationary strategy that is not overtaken by any pure strategy nor by any stationary strategy, under some condition on the transition structure and respectively under genericity. A strategy that is not overtaken by any other strategy, called an overtaking optimal strategy, does not always exist. We provide sufficient conditions for its existence. Next we consider safety objective: the decision maker's goal is to avoid a specific state with the highest possible probability. We argue that the results proven for reachability objective extend to this model. We finally discuss extensions of our results to two-player zero-sum perfect information games.

math.OC↗
arXiv2019

Characterizing the Value Functions of Polynomial Games

We provide a characterization of the set of real-valued functions that can be the value function of some polynomial game. Specifically, we prove that a function $u : \dR \to \dR$ is the value function of some polynomial game if and only if $u$ is a continuous piecewise rational function.

math.OC↗
arXiv2019

Sunspot Equilibrium in General Quitting Games

We prove that positive recursive general quitting games, which are quitting games in which each player may have more than one continue action, admit a sunspot $\ep$-equilibrium, for every $\ep > 0$. To this end we show that the equilibrium set of strategic-form games can be uniformly approximated by a smooth manifold, and develop a new fixed-point theorem for smooth manifolds.

math.PR↗
arXiv2019

Solving Two-State Markov Games with Incomplete Information on One Side *

We study the optimal use of information in Markov games with incomplete information on one side and two states. We provide a finite-stage algorithm for calculating the limit value as the gap between stages goes to 0, and an optimal strategy for the informed player in the limiting game in continuous time. This limiting strategy induces an-optimal strategy for the informed player, provided the gap between stages is small. Our results demonstrate when the informed player should use his information and how.

math.OC↗
arXiv2018

Jointly Controlled Lotteries with Biased Coins

We provide a mechanism that uses two biased coins and implements any distribution on a finite set of elements, in such a way that even if the outcomes of one of the coins is determined by an adversary, the final distribution remains unchanged. We apply this result to show that every quitting game in which at least two players have at least two continue actions has an undiscounted $\ep$-equilibrium, for every $\ep > 0$.

math.PR↗