Nash equilibrium and evolutionary dynamics in semifinalists' dilemma

Seung Ki Baek, Seung-Woo Son, and Hyeong-Chai Jeong

Phys. Rev. E 91, 042144 – Published 30 April 2015

Abstract

We consider a tournament among four equally strong semifinalists. The players have to decide how much stamina to use in the semifinals, provided that the rest is available in the final and the third-place playoff. We investigate optimal strategies for allocating stamina to the successive matches when players' prizes (payoffs) are given according to the tournament results. From the basic assumption that the probability to win a match follows a nondecreasing function of stamina difference, we present symmetric Nash equilibria for general payoff structures. We find three different phases of the Nash equilibria in the payoff space. First, when the champion wins a much bigger payoff than the others, any pure strategy can constitute a Nash equilibrium as long as all four players adopt it in common. Second, when the first two places are much more valuable than the other two, the only Nash equilibrium is such that everyone uses a pure strategy investing all stamina in the semifinal. Third, when the payoff for last place is much smaller than the others, a Nash equilibrium is formed when every player adopts a mixed strategy of using all or none of its stamina in the semifinals. In a limiting case that only last place pays the penalty, this mixed-strategy profile can be proved to be a unique symmetric Nash equilibrium, at least when the winning probability follows a Heaviside step function. Moreover, by using this Heaviside step function, we study the tournament by using evolutionary replicator dynamics to obtain analytic solutions, which reproduces the corresponding Nash equilibria on the population level and gives information on dynamic aspects.

Received 2 May 2014
Revised 21 March 2015

DOI:https://doi.org/10.1103/PhysRevE.91.042144

Authors & Affiliations

Seung Ki Baek^1,*, Seung-Woo Son^2,†, and Hyeong-Chai Jeong^3,‡

¹Department of Physics, Pukyong National University, Busan 608-737, Korea
²Department of Applied Physics, Hanyang University, Ansan 426-791, Korea
³Department of Physics, Sejong University, Seoul 143-747, Korea

^*seungki@pknu.ac.kr
^†sonswoo@hanyang.ac.kr
^‡hcj@sejong.ac.kr

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Vol. 91, Iss. 4 — April 2015

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Images

Figure 1
Schematic phase diagrams with a general payoff structure, where $u$ is the relative payoff of the second-place winner and $v$ is that of the third-place winner, for (a) $h_{0} > 0$ and (b) $h_{0} = 0$ . The region of interest $u \geq v$ is divided into three regions, denoted by I, II, and III, respectively. Region I is characterized by the strategy profile $S^{*} = (x^{*}, x^{*}, x^{*}, x^{*})$ for any pure strategy $x^{*} \in [0, 1]$ and bounded by Eq. (10). However, as drawn (b), it shrinks to a line segment of $0 \leq u = v \leq \frac{1}{2}$ if ${min}_{Δ x > 0} h (Δ x) = 0$ . In region II bounded by Eq. (13), $S^{*}$ is not a Nash equilibrium unless $x^{*} = 1$ . If $v$ is large enough, the players may have a common mixed strategy as a Nash equilibrium in region III (see the text for details).
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Figure 2
Replicator dynamics for (a) $u = v = 0$ and (b) $u = v = 1$ . (a) Among the three fixed points where $\frac{\partial c}{\partial t} = 0$ , the middle one (the open circle) at $c = φ^{- 1}$ is unstable, whereas the other two (the closed circles) are stable. In the middle row, $c$ is shown as a function of $x$ (the thick red line) for a uniform strategy distribution (the thin blue line). It decreases to zero for $x < φ^{- 1}$ , while it increases to one for $x > φ^{- 1}$ , resulting in a step function jumping at $x = φ^{- 1}$ as shown in the bottom row. Hence, $P (x)$ has a sharp peak at $x = φ^{- 1}$ in the steady state. (b) The only stable fixed point is at $c = φ^{- 2}$ . The cumulative probability $c$ decreases if it is larger than $φ^{- 2}$ and increases when it is smaller than $φ^{- 2}$ resulting in the $c (x)$ configuration in the bottom row: It is zero at $x = 0$ , one at $x = 1$ , and $c = φ^{- 2}$ elsewhere, regardless of the initial configuration. Hence, in the steady state, $P (x)$ has two peaks, one at $x = 0$ and the other at $x = 1$ .
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Figure 3
Fraction of population choosing a pure strategy $x$ at time $t$ , obtained by numerical integration of the replicator equation with spatial and temporal step sizes $d x = 10^{- 3}$ and $d t = 10^{- 2}$ . The initial distribution is assumed uniform so that each $x$ has probability $P (x; t = 0) d x = 10^{- 3}$ . (a) The distribution converges toward $x = φ^{- 1}$ (the dotted vertical line) in the winner-take-all case $(u = v = 0)$ . (b) Two peaks emerge at $x = 0$ with weight $φ^{- 2}$ and $x = 1$ with weight $φ^{- 1}$ in the loser-pay-all case $(u = v = 1)$ . Although the simulation is a discretized version of the replicator equation due to the nature of numerical simulation, the results nicely confirm the analytic solutions for continuous strategy space.
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Figure 4
(a) Cumulative $c$ and (b) the corresponding fraction $P$ as a function of $x$ at three different values of $t$ (with $u = v = 0)$ , provided that we start from an initial distribution $P (x; t = 0) = e^{- 4 x} / Z$ with a normalization constant $Z \equiv (1 - e^{- 4}) / 4$ . The curves are obtained by numerical integration of replicator dynamics in the same way as explained in Fig. 3. The vertical dotted lines represent $x = \tilde{x} \approx 0.233$ and the horizontal one $c = φ^{- 1}$ (see the text).
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Figure 5
Cumulative distribution resulting from the Moran process for a finite population of $N$ individuals. We consider $N = 10^{2}$ and $10^{4}$ and assume finite resolution of $d x = 10^{- 3}$ in defining strategy $x$ . Each curve represents a single run with an initial condition sampled from a uniform random distribution of $x$ . (a) In the winner-take-all case $(u = v = 0$ and $t = 10)$ , the cumulative distribution develops a jump near $x \approx 0.6$ for $N = 10^{4}$ , implying a single peak in the strategy distribution around $x \approx 0.6$ , after $O (10)$ time steps. (b) For $u = v = 1, N = 10^{4}$ , and $t = 20$ , the behavior is again consistent with the result of replicator dynamics in that we find a plateau of height $\approx 0.4$ , when $t \sim O (10)$ time steps.
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