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Dynkin ghost games with asymmetry and consolation

Erik Ekström, Yuqiong Wang

Abstract

We study a stopping game of preemption type between two players who both act under uncertain competition. In this framework we introduce, and study the effect of, (i) asymmetry of payoffs, allowing e.g. for different investment costs, and (ii) consolation, i.e. partial compensation to the forestalled stopper. In general, this setting does not offer an explicit equilibrium. Instead, we provide a general verification theorem, which we then use to explore various situations in which a solution can be constructed so that an equilibrium is obtained.

Dynkin ghost games with asymmetry and consolation

Abstract

We study a stopping game of preemption type between two players who both act under uncertain competition. In this framework we introduce, and study the effect of, (i) asymmetry of payoffs, allowing e.g. for different investment costs, and (ii) consolation, i.e. partial compensation to the forestalled stopper. In general, this setting does not offer an explicit equilibrium. Instead, we provide a general verification theorem, which we then use to explore various situations in which a solution can be constructed so that an equilibrium is obtained.

Paper Structure

This paper contains 9 sections, 4 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Literature review
  3. Preview
  4. Problem formulation
  5. A verification theorem
  6. Symmetric games with consolation
  7. Symmetric games with martingale consolation
  8. Symmetric games with supermartingale consolation
  9. Asymmetric games

Key Result

Lemma 2.5

Let $\Gamma^1,\Gamma^2\in\mathcal{A}$, denote by $\gamma_1$ and $\gamma_2$ the corresponding randomised stopping times, and let $\gamma_i(u)$, $i=1,2$ be defined as in gammai. Then and

Figures (1)

  • Figure 1: The boundary $b$ and a simulated path of the reflected process $(\Pi^1,X)$ from Example \ref{['ex1']}. Here $g(x) = (x-3)^+$, $h(x) = (x-4)^+$, $\mu = 0.08$, $\sigma = 0.01$ and $r = 0.1$.

Theorems & Definitions (18)

  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Theorem 3.1
  • proof
  • Remark 4.1
  • ...and 8 more