Simultaneous and sequential choice in a symmetric two-player game with canyon-shaped payoffs

Abstract

This paper investigates a symmetric two-player game with canyon-shaped payoffs, in which a player’s payoff function is smooth and concave above and below the diagonal, but not differentiable on the diagonal. We demonstrate that there exists a first-mover advantage when the two players move sequentially and a player’s preference to the opponent’s choice is monotonic and identical between a higher strategy player and a lower strategy player. We also show that our symmetric two-player game may yield the first-mover advantage outcome in an endogenous timing game with observable delay.

Appendices

Appendix A: The existence of a pure strategy Nash equilibrium with multiple jumps of reaction curves.

The difference $\pi ^{h}(g^{h}(s_{j}),s_{j})-\pi ^{l}(g^{l}(s_{j}),s_{j})$ is positive at $s_{j}=\underline{{c}}$ and negative at $s_{j}=\bar{c}$. We suppose that the jumps across the diagonal of a player’s reaction curve $g(s_{j})$, with $g(\underline{{c}})=g^{h}(\underline{{c}})$ and $g(\bar{c})=g^{l}(\bar{c})$, are finite and also denote $d_{n}$, for $n\in \{1,2,\ldots ,N\}$, as thenth jump point, where $\underline{{c}}<d_{1}<d_{2}<\cdots<d_{N}<\bar{c}$.³¹ The jump point is found by solving $\pi ^{h}(g^{h}(s_{j}),s_{j})-\pi ^{l}(g^{l}(s_{j}),s_{j})=0$. Claim 1 extends the condition that guarantees the existence of a pure strategy Nash equilibrium in Case (ii) with multiple jumps in a player’s reaction curve.

Claim 1

Denote $d_{0}\equiv \underline{{c}}$ and $d_{N+1}\equiv \bar{c}$. If $l,m\in \{0,1,2,\ldots ,N\},$ with l as an even number involving zero, m is an odd number, and $l<m$, such that (1) ($\pi _{1} ^{h}(d_{m},d_{l})>0,$$\pi _{1}^{h}(d_{m+1},d_{l+1})<0)$ and (2) ($\pi _{1} ^{l}(d_{l},d_{m})>0,$$\pi _{1}^{l}(d_{l+1},d_{m+1})<0)$, then a Nash equilibrium $(\hat{s}_{i},\hat{s}_{j})$, for $i,j\in \{a,b\}$ and $i\ne j,$ exists in the set $[d_{l},d_{l+1}]\times [d_{m},d_{m+1}]$ under Case (ii).

Proof

We take Fig. 5 for example, in which there are three jump points in a player’s reaction curve ($N=3)$. By the assumption of l being an even number involving zero, m is an odd number, and $l<m$, we have $g(s_{i})=g^{h}(s_{i})$ for $s_{i}\in [d_{l},d_{l+1}]$ and $g(s_{j})=g^{l}(s_{j})$ for $s_{j}\in [d_{m},d_{m+1}]$. By strict concavity, the assumption of ($\pi _{1}^{h}(d_{m},d_{l})>0,$$\pi _{1} ^{h}(d_{m+1},d_{l+1})<0)$ implies that $g^{h}(d_{l})>d_{m}$ and $g^{h} (d_{l+1})<d_{m+1}$, and the assumption of ($\pi _{1}^{l}(d_{l},d_{m})>0,$$\pi _{1}^{l}(d_{l+1},d_{m+1})<0)$ implies that $g^{l}(d_{m})>d_{l}$ and $g^{l}(d_{m+1})<d_{l+1}$. Monotonicity of $g^{h}$ implies that $g^{h} (s_{i})\in [d_{m},d_{m+1}]$ for all $s_{i}\in [d_{l},d_{l+1}]$, and monotonicity of $g^{l}$ implies that $g^{l}(s_{j})\in [d_{l} ,d_{l+1}]$ for all $s_{j}\in [d_{m},d_{m+1}]$. The two reaction curves, $g^{l}(s_{j})$ and $g^{h}(s_{i})$, must intersect in the set $[d_{l} ,d_{l+1}]\times [d_{m},d_{m+1}]$. $\square$

Fig. 5

The existence of a pure strategy Nash equilibrium with multiple jumps of reaction curves under Case (ii)

Claim 1 states that if the reaction curves are restricted to certain compact subsets of the joint strategy space, $[d_{l},d_{l+1}]\times [d_{m},d_{m+1}]$, then the existence of a Nash equilibrium in Case (ii) is guaranteed. If condition (2) in Claim 1 is replaced by ($\pi _{1}^{l} (d_{l+1},d_{m})<0,$$\pi _{1}^{l}(d_{l},d_{m+1})>0)$, then a Nash equilibrium $(\hat{s}_{i},\hat{s}_{j})$ exists in the set $[d_{l},d_{l+1}]\times [d_{m},d_{m+1}]$ under Case (iii). Similarly, if condition (1) in Claim 1 is replaced by ($\pi _{1}^{h}(d_{m+1},d_{l})<0,$$\pi _{1}^{h}(d_{m},d_{l+1})>0)$, then a Nash equilibrium $(\hat{s}_{i},\hat{s}_{j})$ exists in the set $[d_{l},d_{l+1}]\times [d_{m},d_{m+1}]$ under Case (iv).

Appendix B

Lemma 3

Suppose$\pi _{2}^{l}(s_{i},s_{j})<0$and$\pi _{2}^{h} (s_{i},s_{j})<0$.

• (i) If$\pi _{12}^{h}<0$and$\pi _{12}^{l}<0$, then$\hat{s}_{a} ^{l}<s_{a}^{l}{}^{*}<d<s_{b}^{h}{}^{*}<\hat{s}_{b}^{h}$and$s_{b}^{l} {}^{*}<\hat{s}_{b}^{l}<d<\hat{s}_{a}^{h}<s_{a}^{h}{}^{*}$.

• (ii) If$\pi _{12}^{h}>0$and$\pi _{12}^{l}>0$, then$s_{a}^{l} {}^{*}<\hat{s}_{a}^{l}<d<s_{b}^{h}{}^{*}<\hat{s}_{b}^{h}$and$s_{b} ^{l}{}^{*}<\hat{s}_{b}^{l}<d<s_{a}^{h}{}^{*}<\hat{s}_{a}^{h}$.

• (iii) If$\pi _{12}^{h}>0$and$\pi _{12}^{l}<0$, then$s_{a}^{l} {}^{*}<\hat{s}_{a}^{l}<d<s_{b}^{h}{}^{*}<\hat{s}_{b}^{h}$and$s_{b} ^{l}{}^{*}<\hat{s}_{b}^{l}<d<\hat{s}_{a}^{h}<s_{a}^{h}{}^{*}$.

• (iv) If$\pi _{12}^{h}<0$and$\pi _{12}^{l}>0$, then$\hat{s}_{a} ^{l}<s_{a}^{l}{}^{*}<d<s_{b}^{h}{}^{*}<\hat{s}_{b}^{h}$and$s_{b}^{l} {}^{*}<\hat{s}_{b}^{l}<d<s_{a}^{h}{}^{*}<\hat{s}_{a}^{h}$.

Proof

Suppose $s_{a}^{*}\le s_{b}^{*}$. In Cases (i) and (iv), since $\pi _{1}^{l}(s_{a},g^{h}(s_{a}))+\pi _{2}^{l}(s_{a},s_{b})\cdot {g}^{\prime }(s_{a})>\pi _{1}^{l}(s_{a},g^{h}(s_{a}))$ for all $s_{a}\le s_{b}$, the leader chooses a higher choice in a sequential game versus that in a simultaneous game, $s_{a}^{l}{}^{*}>\hat{s}_{a}^{l}$, which follows that $s_{b}^{h}{}^{*}<\hat{s}_{b}^{h}$ under strategic substitutes. In Cases (ii) and (iii), since $\pi _{1}^{l}(s_{a},g^{h}(s_{a}))+\pi _{2}^{l}(s_{a} ,s_{b})\cdot {g}^{\prime }(s_{a})<\pi _{1}^{l}(s_{a},g^{h}(s_{a}))$ for all $s_{a}\le s_{b}$, the leader chooses a lower choice in a sequential game versus that in a simultaneous game, $s_{a}^{l}{}^{*}<\hat{s}_{a}^{l}$, which follows that $s_{b}^{h}{}^{*}<\hat{s}_{b}^{h}$ under strategic complements.

Suppose that $s_{a}^{*}\ge s_{b}^{*}$. In Cases (i) and (iii) since $\pi _{1}^{h}(s_{a},g^{l}(s_{a}))+\pi _{2}^{h}(s_{a},s_{b})\cdot {g}^{\prime } (s_{a})>\pi _{1}^{h}(s_{a},g^{l}(s_{a}))$ for all $s_{a}\ge s_{b}$, the leader chooses a higher choice in a sequential game versus that in a simultaneous game, $s_{a}^{h}{}^{*}>\hat{s}_{a}^{h}$, which follows that $s_{b}^{l} {}^{*}<\hat{s}_{b}^{l}$ under strategic substitutes. In Cases (ii) and (iv), since $\pi _{1}^{h}(s_{a},g^{l}(s_{a}))+\pi _{2}^{h}(s_{a},s_{b})\cdot {g}^{\prime }(s_{a})<\pi _{1}^{h}(s_{a},g^{l}(s_{a}))$ for all $s_{a}\ge s_{b}$, the leader chooses a lower choice in a sequential game versus that in a simultaneous game, $s_{a}^{h}{}^{*}<\hat{s}_{a}^{h}$, which follows that $s_{b}^{l}{}^{*}<\hat{s}_{b}^{l}$ under strategic complements. $\square$

Lemma 4

Suppose$\pi _{2}^{l}(s_{i},s_{j})>0$and$\pi _{2}^{h}(s_{i},s_{j})>0$.

• (i) If$\pi _{12}^{h}<0$and$\pi _{12}^{l}<0$, then$s_{a}^{l}{} ^{*}<\hat{s}_{a}^{l}<d<\hat{s}_{b}^{h}<s_{b}^{h}{}^{*}$and$\hat{s} _{b}^{l}<s_{b}^{l}{}^{*}<d<s_{a}^{h}{}^{*}<\hat{s}_{a}^{h}$.

• (ii) If$\pi _{12}^{h}>0$and$\pi _{12}^{l}>0$, then$\hat{s}_{a} ^{l}<s_{a}^{l}{}^{*}<d<\hat{s}_{b}^{h}<s_{b}^{h}{}^{*}$and$\hat{s} _{b}^{l}<s_{b}^{l}{}^{*}<d<\hat{s}_{a}^{h}<s_{a}^{h}{}^{*}$.

• (iii) If$\pi _{12}^{h}>0$and$\pi _{12}^{l}<0$, then$\hat{s}_{a} ^{l}<s_{a}^{l}{}^{*}<d<\hat{s}_{b}^{h}<s_{b}^{h}{}^{*}$and$\hat{s} _{b}^{l}<s_{b}^{l}{}^{*}<d<s_{a}^{h}{}^{*}<\hat{s}_{a}^{h}$.

• (iv) If$\pi _{12}^{h}<0$and$\pi _{12}^{l}>0$, then$s_{a}^{l} {}^{*}<\hat{s}_{a}^{l}<d<\hat{s}_{b}^{h}<s_{b}^{h}{}^{*}$and$\hat{s}_{b}^{l}<s_{b}^{l}{}^{*}<d<\hat{s}_{a}^{h}<s_{a}^{h}{}^{*}$.

Proof

Lemma 4 is the reverse of the order on a player’s strategy space in Lemma 3. In particular, letting $s_{i}^{\prime }=-s_{i}$ and swapping h and l, Lemmas 4 (i), (ii), (iii), and (iv) follow from Lemmas 3 (i), (ii), (iv), and (iii), respectively. $\square$

Lemma 5

Suppose$\pi _{2}^{l}(s_{i},s_{j})>0$and$\pi _{2}^{h}(s_{i},s_{j})<0$.

• (i) If$\pi _{12}^{h}<0$ and $\pi _{12}^{l}<0$, then$s_{a}^{l}{} ^{*}<g^{l}(s_{b}^{h}{}^{*})<\hat{s}_{a}^{l}<d<\hat{s}_{b}^{h}<s_{b} ^{h}{}^{*}$and$s_{b}^{l}{}^{*}<\hat{s}_{b}^{l}<d<\hat{s}_{a}^{h}<g^{h}(s_{b}^{l}{}^{*})<s_{a}^{h}{}^{*}$.

• (ii) If$\pi _{12}^{h}>0$and$\pi _{12}^{l}>0$, then$\hat{s}_{a} ^{l}<g^{l}(s_{b}^{h}{}^{*})<s_{a}^{l}{}^{*}<d<\hat{s}_{b}^{h}<s_{b} ^{h}{}^{*}$and$s_{b}^{l}{}^{*}<\hat{s}_{b}^{l}<d<s_{a}^{h}{}^{*}<g^{h}(s_{b}^{l}{}^{*})<\hat{s}_{a}^{h}$.

• (iii) If$\pi _{12}^{h}>0$and$\pi _{12}^{l}<0$, then$g^{l}(s_{b} ^{h}{}^{*})<\hat{s}_{a}^{l}<s_{a}^{l}{}^{*}<d<\hat{s}_{b}^{h}<s_{b} ^{h}{}^{*}$and$s_{b}^{l}{}^{*}<\hat{s}_{b}^{l}<d<g^{h}(s_{b}^{l} {}^{*})<\hat{s}_{a}^{h}<s_{a}^{h}{}^{*}$.

• (iv) If$\pi _{12}^{h}<0$and$\pi _{12}^{l}>0$, then$s_{a}^{l} {}^{*}<\hat{s}_{a}^{l}<g^{l}(s_{b}^{h}{}^{*})<d<\hat{s}_{b}^{h} <s_{b}^{h}{}^{*}$and$s_{b}^{l}{}^{*}<\hat{s}_{b}^{l}<d<s_{a}^{h} {}^{*}<\hat{s}_{a}^{h}<g^{h}(s_{b}^{l}{}^{*})$.

Proof

In Case (i) if $s_{a}^{*}\le s_{b}^{*}$, then we can show the leader chooses a lower choice in a sequential game versus that in a simultaneous game, $s_{a}^{l}{}^{*}<\hat{s}_{a}^{l}$, and $s_{b}^{h} {}^{*}>\hat{s}_{b}^{h}$ and $g^{l}(s_{b}^{h}{}^{*})<\hat{s}_{a}^{l}$ under strategic substitutes. We then claim that $g^{l}(s_{b}^{h}{}^{*})>s_{a}^{l}{}^{*}$. Assume for a moment that this is not true—that is, $g^{l}(s_{b}^{h}{}^{*})\le s_{a}^{l}{}^{*}$. Thus, $\pi _{1}^{l} (s_{a}^{l}{}^{*},s_{b}^{h}{}^{*})\le \pi _{1}^{l}(g^{l}(s_{b}^{h}{} ^{*}),s_{b}^{h}{}^{*})=0$. The first inequality follows since $\pi _{11}^{l}<0$, and the second equality follows from the definition of $g^{l}$. Since $\pi _{1}^{l}(s_{a}^{l}{}^{*},s_{b}^{h}{}^{*})\le 0$ and $\pi _{12}^{h}(s_{b}^{h}{}^{*},s_{a}^{l}{}^{*})<0$, $\pi _{2}^{l}(s_{a}^{l} {}^{*},s_{b}^{h}{}^{*})\le 0$ from Eq. (2), which is a contradiction.

The leader conversely chooses a higher choice in a sequential game versus that in a simultaneous game, $s_{a}^{h}{}^{*}>\hat{s}_{a}^{h}$, if $s_{a}^{*}\ge s_{b}^{*}$. It follows that $s_{b}^{l}{}^{*}<\hat{s}_{b}^{l}$ and $g^{h}(s_{b}^{l}{}^{*})>\hat{s}_{a}^{h}$ under strategic substitutes. We then claim that $g^{h}(s_{b}^{l}{}^{*})<s_{a}^{h}{}^{*}$. Assume that this is not true—that is, $g^{h}(s_{b}^{l}{}^{*})\ge s_{a}^{h}{}^{*}$. Thus, $\pi _{1}^{h}(s_{a}^{h}{}^{*},s_{b}^{l}{}^{*})\ge \pi _{1} ^{h}(g^{h}(s_{b}^{l}{}^{*}),s_{b}^{l}{}^{*})=0$. The first inequality follows, since $\pi _{11}^{h}<0$, and the second equality follows from the definition of $g^{h}$. Since $\pi _{1}^{h}(s_{a}^{h}{}^{*},s_{b}^{l}{} ^{*})\ge 0$ and $\pi _{12}^{l}(s_{b}^{l}{}^{*},s_{a}^{h}{}^{*})<0$, $\pi _{2}^{h}(s_{a}^{h}{}^{*},s_{b}^{l}{}^{*})\ge 0$ from Eq. (2), which is a contradiction. The proofs of Lemmas 5 (ii), (iii), and (iv) are similar to those of Lemma 5 (i), for which we exclude the former in the paper to save space. $\square$