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Nash Equilibria in the Showcase Showdown game with unlimited spins

L. Bayón, P. Fortuny ayuso, J. M. Grau, A. M. Oller-Marcén, M. M Ruíz

TL;DR

The paper addresses Nash equilibria for the n-player Showcase Showdown with unlimited spins under sequential (complete information) and three no-information payoff variants. It develops a threshold-based framework where each player's optimal action is governed by a greed threshold, derives explicit threshold sequences (θ_n, α_n, γ_n, ε_n, δ_n) through integral equations and fixed-point arguments, and provides recursive expressions for winning probabilities. It also examines coalitions in the sequential game and enumerates equilibria for the no-information variants, including numerical values up to n=10 and several asymptotic insights. The results advance understanding of optimal stopping in multi-agent settings, reveal how payoff structures influence equilibria, and suggest directions for future theoretical and applied work in sequential decision games with uncertainty.

Abstract

The game of \emph{Showcase Showdown} with unlimited spins is investigated as an $n$-players continuous game, and the Nash Equilibrium strategies for the players are obtained. The sequential game with information on the results of the previous players is studied, as well as three variants: no information, possibility of draw, and different modalities of winner payoff.

Nash Equilibria in the Showcase Showdown game with unlimited spins

TL;DR

The paper addresses Nash equilibria for the n-player Showcase Showdown with unlimited spins under sequential (complete information) and three no-information payoff variants. It develops a threshold-based framework where each player's optimal action is governed by a greed threshold, derives explicit threshold sequences (θ_n, α_n, γ_n, ε_n, δ_n) through integral equations and fixed-point arguments, and provides recursive expressions for winning probabilities. It also examines coalitions in the sequential game and enumerates equilibria for the no-information variants, including numerical values up to n=10 and several asymptotic insights. The results advance understanding of optimal stopping in multi-agent settings, reveal how payoff structures influence equilibria, and suggest directions for future theoretical and applied work in sequential decision games with uncertainty.

Abstract

The game of \emph{Showcase Showdown} with unlimited spins is investigated as an -players continuous game, and the Nash Equilibrium strategies for the players are obtained. The sequential game with information on the results of the previous players is studied, as well as three variants: no information, possibility of draw, and different modalities of winner payoff.

Paper Structure

This paper contains 13 sections, 18 theorems, 108 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The Showcase Showdown game
  3. Optimal strategy. Threshold strategy.
  4. Cumulative distribution function of the score.
  5. Game I. Sequential game
  6. Coalition between first and second player
  7. Calculation of $\theta_{2}{}(x)$
  8. Coalition between first and third player
  9. Game II --- No-information game
  10. Game II.1: Non-constant sum
  11. Game II.2: Zero-sum game
  12. Game II.3: Non-symmetric and constant-sum
  13. Conclusions and future perspectives

Key Result

Lemma 1

Let $h:[0,1]\rightarrow\mathbb{R}$ be a non-decreasing monotone function, and let $\tilde{h}(x)=h(0) x + \int_x^1h(t)dt$, for $x\in[0,1]$. Define Then the function is a non-increasing monotone $C[0,1]$ function, and the only one satisfying the integral equation

Figures (3)

  • Figure 1: Winning probability of Player 1 (blue) depending on the threshold of Player $2$ ($y$). In orange: winning probability of each player in the Nash equilibrium.
  • Figure 2: Payoff $z=\mathbb{P}_{1}(\gamma_{3},x,y)$. In blue, $z=0$: notice how $\mathbb{P}_1(\gamma_3,x,y)\geq 0$ everywhere.
  • Figure 3: Solutions of Equation \ref{['eq:90']} for $n=2,3,\ldots, 6$. The decreasing plots correspond to a) in \ref{['eq:90']}, while the increasing ones to b).

Theorems & Definitions (48)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Example 1
  • Example 2
  • Lemma 2
  • proof
  • Remark 1
  • Corollary 1
  • ...and 38 more