arXiv subjects
Publications and source records attributed to Mehmet Mars Seven.
We present several models of strategic choice concerning equilibrium reasoning and behaviour, as well as new applications to dynamic games. We propose concepts such as historical equilibrium, simple Nash equilibrium, cautious backward induction, and an approach inspired by how chess players reason in sequential games.
We study whether zero-sum decision rules, maximin and minimax, harm agents' interests in positive-sum games relative to Nash equilibrium behaviour or, more generally, than best response behaviour. Contrary to an influential evolutionary view, we give illustrations where maximin serves an agent's interests better than Nash equilibrium behaviour. Two new sets of results show that these illustrations are not idiosyncratic. First, for any selected Nash equilibrium in cardinal games, we construct a strategically equivalent game where a maximin profile yields the same pay-offs and we fully characterise when maximin can be made to Pareto dominate that Nash equilibrium and the entire Nash equilibrium set. Second, we show that the relevant Maximin Pareto dominance classes are not knife-edge: they are generically interior, and strict maximin profile dominance occurs on a non-empty open set if and only if a player has at least two actions and the other has at least three.
In this paper, we propose a game-theoretic model to study the properties of the worst-case regret of the greedy strategy under complete (Knightian) uncertainty. In a game between a decision-maker (DM) and an adversarial agent (Nature), Nature chooses an unknown state determining the distribution of ratings for each product. The DM observes a realization of product ratings and then chooses a product according to a strategy. For arbitrary numbers of products and ratings, we first study the equal-observations case in which every product has the same number of observations. In this benchmark, we establish matching upper and lower bounds on the worst-case regret, showing that the regret vanishes as the number of observations increases and that the greedy strategy is rate-optimal up to universal constants. In the special case with two products and two ratings, we show that with one observation per product the greedy strategy is minimax-optimal with respect to worst-case regret. We then allow products to have different numbers of observations. Greedy remains robust in a conservative sense: its worst-case regret is controlled by the least-reviewed product. However, unequal numbers of observations can also change greedy's exact worst-case behavior. In particular, adding observations for only one product can increase greedy's worst-case regret. Finally, we test the model on data collected from Google reviews for restaurants, showing that the greedy strategy's empirical performance closely aligns with the theoretical findings.
We extend the optimin notion of Ismail (2025) from mixed strategy profiles to correlated distributions. A correlated distribution is evaluated by the worst expected payoff each player can receive when opponents may either obey their private recommendations or make unilateral recommendation-contingent deviations that are strictly profitable under the posterior induced by the distribution. Correlated optimins are Pareto optimal with respect to this vector of guaranteed payoffs. We show that a correlated optimin exists in every finite game. In addition, for every correlated equilibrium, there exists a correlated optimin such that every player's guaranteed payoff is weakly higher than his or her correlated equilibrium payoff. In two-player zero-sum games, correlated optimin coincides with correlated equilibrium and yields the maximin value. Outside zero-sum games, correlated optimin may strictly improve upon all correlated equilibria. We illustrate this with a simple 2x2 game with a unique correlated and coarse correlated equilibrium, in which there exists a correlated optimin that strictly Pareto dominates the equilibrium payoff.
Many environments assign several Elo ratings to the same agent: a chess player has classical, rapid, and blitz ratings; an online platform may rate by time control, mode, or format; an evaluator may rate performance across tasks or roles. This paper axiomatizes when such a vector of ratings can be reduced to a single scalar rating that is itself on the Elo scale. We impose three substantive conditions: same-scale normalization (a uniform profile keeps its rating), recursive consistency (aggregating in blocks gives the same answer as aggregating directly, provided each block carries the total weight of its members), and marginal Elo-strength consistency (for two equally weighted coordinates, the ratio of marginal contributions to the combined rating equals the ordinary Elo odds). The unique rating rule satisfying these conditions converts each component to its Elo strength, takes a weighted arithmetic mean of strengths, and converts back. We show how this rule differs from a random-format lottery and from rating-scale averaging, prove the axioms are independent, and illustrate the rule on combining classical, rapid, and blitz ratings.
We propose the Subgame Credible Nash Equilibrium (SCNE), a refinement of subgame perfect Nash equilibrium (SPNE) for multi-stage games. SCNE retains the internal credibility requirement of SPNE -- equilibrium behavior in every subgame -- and adds an external credibility requirement across equivalent subgames: whenever a player's prescribed continuation strategy differs across equivalent subgames, her own continuation payoff must not decrease. The intuition is that credible punishments or promises should not strictly harm the punisher relative to an equivalent no-punishment subgame. The SCNE eliminates equilibria sustained by self-harming punishments or promises while preserving existence. Every multi-stage game admits an SCNE, and if each stage game has a unique Nash equilibrium, the SCNE is unique.
In this note, we introduce a novel performance rating system called Performance Rating Equilibrium (PRE). A PRE is a vector of ratings for each player, such that if these ratings were each player's initial rating at the start of a tournament, scoring the same points against the same opponents would leave each player's initial rating unchanged. In other words, all players' initial ratings perfectly predict their actual scores in the tournament. This property, however, does not hold for the well-known Tournament Performance Rating. PRE is defined as a fixed point of a multidimensional rating function. We show that such a fixed point, and hence a PRE, exists under mild conditions. We provide an implementation of PRE along with several empirical applications. PREs have broad applicability, from sports competitions to the evaluation of large language models.
In this paper, I formalize intelligence measurement in games by introducing mechanisms that assign a real number -- interpreted as an intelligence score -- to each player in a game. This score quantifies the ex-post strategic ability of the players based on empirically observable information, such as the actions of the players, the game's outcome, strength of the players, and a reference oracle machine such as a chess-playing artificial intelligence system. Specifically, I introduce two main concepts: first, the Game Intelligence (GI) mechanism, which quantifies a player's intelligence in a game by considering not only the game's outcome but also the "mistakes" made during the game according to the reference machine's intelligence. Second, I define gamingproofness, a practical and computational concept of strategyproofness. To illustrate the GI mechanism, I apply it to an extensive dataset comprising over a billion chess moves, including over a million moves made by top 20 grandmasters in history. Notably, Magnus Carlsen emerges with the highest GI score among all world championship games included in the dataset. In machine-vs-machine games, the well-known chess engine Stockfish comes out on top.