Sample Out-of-Sample Inference Based on Wasserstein Distance

Financial institutions make decisions according to a model of uncertainty. At the same time, regulators often evaluate the risk exposure of these institutions using a model of uncertainty, which is often different from the one used by the institutions. How can one incorporate both views into a single framework? This paper provides such a framework. It quantifies the impact of the misspecification inherent to the financial institution data-driven model via the introduction of an adversarial player. The adversary replaces the institution's generated scenarios by the regulator's scenarios subject to a budget constraint and a cost that measures the distance between the two sets of scenarios (using what in statistics is known as the Wasserstein distance). This paper also harnesses statistical theory to make inference about the size of the estimated error when the sample sizes (both of the institution and the regulator) are large. The framework is explained more broadly in the context of distributionally robust optimization (a class of perfect information games, in which decisions are taken against an adversary that perturbs a baseline distribution).

Correction to: Perfect information games where each player acts only once

In the original publication of the article, the last name of the corresponding author was omitted by mistake. The correct name should read: P. Jean-Jacques Herings. The original article has been corrected.

Perfect information games where each player acts only once

We study perfect information games played by an infinite sequence of players, each acting only once in the course of the game. We introduce a class of frequency-based minority games and show that these games have no subgame perfect $$\epsilon $$ ϵ -equilibrium for any $$\epsilon $$ ϵ sufficiently small. Furthermore, we present a number of sufficient conditions to guarantee existence of subgame perfect $$\epsilon $$ ϵ -equilibrium.

Waiter–Client and Client–Waiter Colourability and $k$–SAT Games

Waiter–Client and Client–Waiter games are two–player, perfect information games, with no chance moves, played on a finite set (board) with special subsets known as the winning sets. Each round of the biased $(1:q)$ Waiter–Client game begins with Waiter offering $q+1$ previously unclaimed elements of the board to Client, who claims one and leaves the remaining $q$ elements to be claimed by Waiter immediately afterwards. In a $(1:q)$ Client–Waiter game, play occurs in the same way except in each round, Waiter offers $t$ elements for any $t$ in the range $1\leqslant t\leqslant q+1$. If Client fully claims a winning set by the time all board elements have been offered, he wins in the Client–Waiter game and loses in the Waiter–Client game. We give an estimate for the threshold bias (i.e. the unique value of $q$ at which the winner of a $(1:q)$ game changes) of the $(1:q)$ Waiter–Client and Client–Waiter versions of two different games: the non–2–colourability game, played on the edge set of a complete $k$–uniform hypergraph, and the $k$–SAT game. More precisely, we show that the threshold bias for the Waiter–Client and Client–Waiter versions of the non–2–colourability game is $\frac{1}{n}\binom{n}{k}2^{\mathcal{O}_k(k)}$ and $\frac{1}{n}\binom{n}{k}2^{-k(1+o_k(1))}$ respectively. Additionally, we show that the threshold bias for the Waiter–Client and Client–Waiter versions of the $k$–SAT game is $\frac{1}{n}\binom{n}{k}$ up to a factor that is exponential and polynomial in $k$ respectively. This shows that these games exhibit the probabilistic intuition.