Quantal response equilibrium in a double auction

Abstract

This paper establishes existence and uniqueness of Quantal Response Equilibrium (QRE) in a double auction. The concept of QRE has the intuitive property that a deviation from best response is less likely the higher the cost associated with the deviation itself. Thanks to such property, the QRE accommodates stochastic elements in the analysis of the strategic decision-making that arises in the double auction. By providing a theoretical alternative to the Bayesian Nash Equilibrium model, the QRE model offers an appealing tool for analyzing data of double auction experiments.

Appendices

Appendix 1

Consider a double auction with $m$ buyers and $n$ sellers. Denote the density functions of the bids and the asks with $g_{b}$ and $g_{s}$, respectively, and the cumulative density functions with $G_{b}$ and $G_{s}$, respectively. The formula for a buyer’s expected payoff [Eq. (1)] employs the formula of the joint density of the $m$th and ($m+1$)th order statistics of the $m+n-1$ bids and asks, $f^B_{(m),(m+1)}$. Denote $k=m$. Then:

$$\begin{aligned}&f^B_{(k),(k+1)} (x,y) = \\&n (n-1) g_{s}(x) g_{s}(y) \sum _{\begin{array}{c} i + j = k-1 \\ 0 \le i \le m-1 \\ 0 \le j \le n-2 \end{array} } \left( {\begin{array}{c}m-1\\ i\end{array}}\right) \left( {\begin{array}{c}n-2\\ j\end{array}}\right) G_{b}(x)^{i} G_{s}(x)^{j} ( 1 - G_{b} (y) ) ^{m-1-i}\\&\times ( 1 - G_{s} (y) ) ^{n-2-j} \\&+\, n (m\!-\!1) g_{s}(x) g_{b}(y) \sum _{\begin{array}{c} i + j = k-1 \\ 0 \le i \le m-2 \\ 0 \le j \le n-1 \end{array} } \left( {\begin{array}{c}m\!-\!2\\ i\end{array}}\right) \left( {\begin{array}{c}n\!-\!1\\ j\end{array}}\right) G_{b}(x)^{i} G_{s}(x)^{j} ( 1 - G_{b} (y) ) ^{m-2-i}\\&\times ( 1 - G_{s} (y) ) ^{n-1-j} \\&+\, (m-1) n g_{b}(x) g_{s}(y) \sum _{\begin{array}{c} i + j = k-1 \\ 0 \le i \le m-2 \\ 0 \le j \le n-1 \end{array} } \left( {\begin{array}{c}m\!-\!2\\ i\end{array}}\right) \left( {\begin{array}{c}n\!-\!1\\ j\end{array}}\right) G_{b}(x)^{i} G_{s}(x)^{j} ( 1 \!-\! G_{b} (y) ) ^{m-2-i}\\&\times ( 1 - G_{s} (y) ) ^{n-1-j} \\&+\, (m\!-\!1) (m\!-\!2) g_{b}(x) g_{b}(y) \sum _{\begin{array}{c} i + j = k-1 \\ 0 \le i \le m\!-\!3 \\ 0 \le j \le n \end{array} } \left( {\begin{array}{c}m\!-\!3\\ i\end{array}}\right) \left( {\begin{array}{c}n\\ j\end{array}}\right) G_{b}(x)^{i} G_{s}(x)^{j} ( 1 \!-\! G_{b} (y) ) ^{m-3-i}\\&\times ( 1 - G_{s} (y) ) ^{n-j} \end{aligned}$$

The formula for a seller’s expected payoff [Eq. (2)] employs the formula of the joint density of the $(m-1)$th and $m$th order statistics of the $m+n-1$ bids and asks, $f^S_{(m-1),(m)}$. Denote $k=m-1$. Then:

$$\begin{aligned}&f^S_{(k),(k+1)} (x,y) = \\&(n-1) (n-2) g_{s}(x) g_{s}(y) \sum _{\begin{array}{c} i + j = k-1 \\ 0 \le i \le m \\ 0 \le j \le n-3 \end{array} } \left( {\begin{array}{c}m\\ i\end{array}}\right) \left( {\begin{array}{c}n-3\\ j\end{array}}\right) G_{b}(x)^{i} G_{s}(x)^{j} ( 1 - G_{b} (y) ) ^{m-i}\\&\times ( 1 - G_{s} (y) ) ^{n-3-j} \\&+\, (n\!-\!1) m g_{s}(x) g_{b}(y) \sum _{\begin{array}{c} i + j = k-1 \\ 0 \le i \le m-1 \\ 0 \le j \le n-2 \end{array} } \left( {\begin{array}{c}m-1\\ i\end{array}}\right) \left( {\begin{array}{c}n\!-\!2\\ j\end{array}}\right) G_{b}(x)^{i} G_{s}(x)^{j} ( 1 - G_{b} (y) ) ^{m-1-i}\\&\times ( 1 - G_{s} (y) ) ^{n-2-j} \\&+\, m (n\!-\!1) g_{b}(x) g_{s}(y) \sum _{\begin{array}{c} i + j = k-1 \\ 0 \le i \le m-1 \\ 0 \le j \le n-2 \end{array} } \left( {\begin{array}{c}m-1\\ i\end{array}}\right) \left( {\begin{array}{c}n-2\\ j\end{array}}\right) G_{b}(x)^{i} G_{s}(x)^{j} ( 1 - G_{b} (y) ) ^{m-1-i}\\&\times ( 1 - G_{s} (y) ) ^{n-2-j} \\&+\, m (m\!-\!1) g_{b}(x) g_{b}(y) \sum _{\begin{array}{c} i + j = k-1 \\ 0 \le i \le m-2 \\ 0 \le j \le n-1 \end{array} } \left( {\begin{array}{c}m\!-\!2\\ i\end{array}}\right) \left( {\begin{array}{c}n\!-\!1\\ j\end{array}}\right) G_{b}(x)^{i} G_{s}(x)^{j} ( 1 - G_{b} (y) ) ^{m-2-i}\\&\times ( 1 - G_{s} (y) ) ^{n-1-j} \end{aligned}$$

Appendix 2

By differentiating Eqs. (3) and (4) with respect to $y$, we obtain two differential equations in the choice densities:

$$\begin{aligned} q'_{B} ( y ) = \lambda \pi '_{B}( y ) q_{B}(y)\end{aligned}$$

(24)

$$\begin{aligned} q'_{S} ( y ) = \lambda \pi '_{S}( y ) q_{S}(y) \end{aligned}$$

(25)

We obtain $\pi '_{B}( y )$ and $\pi '_{S}( y )$ by differentiating the payoff functions $\pi _{B}$ and $\pi _{S}$ with respect to bid $b$ and ask $a$, respectively, and evaluating $\pi '_{B}$ and $\pi '_{S}$ at $b=y$ and $a=y$, respectively. For the buyer, differentiating $\pi _{B}$ produces:

$$\begin{aligned} \pi '_{B}( y )&= - \int \limits _{\underline{z}}^{y} \left( v - [(1-k)s + k y ] \right) f^{B}_{(m),(m+1)}(s, y) \mathrm{d} s \nonumber \\&+ \int \limits _{y}^{\overline{z}} \left( v - [(1-k)y + k y ] \right) f^{B}_{(m),(m+1)}(y, t) \mathrm{d}t \nonumber \\&- k \int \limits _{y}^{\overline{z}} \int \limits _{\underline{z}}^{y} f^{B}_{(m),(m+1)}(s,t) \mathrm{d}s \mathrm{d} t \nonumber \\&+ \int \limits _{\underline{z}}^{y} \left( v - [(1-k)s + k y ] \right) f^{B}_{(m),(m+1)}(s, y) \mathrm{d}s \end{aligned}$$

which simplifies to:

$$\begin{aligned} \pi '_{B}( y )&= \left( v - y \right) \int \limits _{y}^{\overline{z}} f^{B}_{(m),(m+1)}(y, t) \mathrm{d} t - k \int \limits _{y}^{\overline{z}} \int \limits _{\underline{z}}^{y} f^{B}_{(m),(m+1)}(s,t) \mathrm{d}s \mathrm{d}t\nonumber \\&= \left( v - y \right) f^{B}_{(m)}(y) - k M (y) {} \end{aligned}$$

(26)

In (26) $f^{B}_{(m)}(y)$ is the density of $\zeta _{(m)}$ evaluated at a bid $b=y$:

$$\begin{aligned} f^{B}_{(m)}(y) = n K(y) g_{s}(y) + (m-1) L(y) g_{b}(y) \end{aligned}$$

(27)

where:

$$\begin{aligned} K(y) \!&= \! \sum _{\begin{array}{c} i + j = m\!-\!1 \\ 0 \le i \le m\!-\!1 \\ 0 \le j \le n\!-\!1 \end{array} } \left( {\begin{array}{c}m\!-\!1\\ i\end{array}}\right) \left( {\begin{array}{c}n\!-\!1\\ j\end{array}}\right) G_{b}(y)^{i} G_{s}(y)^{j} ( 1 \!-\! G_{b} (y) ) ^{m-1-i} ( 1 \!-\! G_{s} (y) ) ^{n-1-j}\\ L(y)&= \sum _{\begin{array}{c} i + j = m-1 \\ 0 \le i \le m-2 \\ 0 \le j \le n \end{array} } \left( {\begin{array}{c}m-2\\ i\end{array}}\right) \left( {\begin{array}{c}n\\ j\end{array}}\right) G_{b}(y)^{i} G_{s}(y)^{j} ( 1 - G_{b} (y) ) ^{m-2-i} ( 1 - G_{s} (y) ) ^{n-j} \end{aligned}$$

In (26) $M (y)$ corresponds to:

$$\begin{aligned} M (y)&= \int \limits _{y}^{\overline{z}} \int \limits _{\underline{z}}^{y} f^{B}_{(m),(m+1)}(s,t) \mathrm{d}s \mathrm{d}t = P(\zeta _{(m)} < b=y < \zeta _{(m+1)})\\&= \sum _{\begin{array}{c} i + j = m \\ 0 \le i \le m-1 \\ 0 \le j \le n \end{array} } \left( {\begin{array}{c}m-1\\ i\end{array}}\right) \left( {\begin{array}{c}n\\ j\end{array}}\right) G_{b}(y)^{i} G_{s}(y)^{j} ( 1 - G_{b} (y) ) ^{m-1-i} ( 1 - G_{s} (y) ) ^{n-j} \end{aligned}$$

Therefore:

$$\begin{aligned} \pi '_{B}( y ) = \left( v - y \right) \left( n K(y) g_{s}(y) + (m-1) L(y) g_{b}(y) \right) - k M(y) \end{aligned}$$

(28)

For the seller, differentiating $\pi _{S}$ produces:

$$\begin{aligned} \pi '_{S}( y )&= - \int \limits _{\underline{z}}^{y} \left( (1-k)y + k y - c \right) f^{S}_{(m-1),(m) } (s,y) \mathrm{d}s \nonumber \\&{} + \int \limits _{y}^{\overline{z}} \left( (1-k)y + k t - c \right) f^{S}_{(m-1),(m) } (y, t ) \mathrm{d} t \nonumber \\&{} + (1-k) \int \limits _{y}^{\overline{z}} \int \limits _{\underline{z}}^{y} f^{S}_{(m-1),(m)} (s,t) \mathrm{d}s \mathrm{d}t \nonumber \\&{} - \int \limits _{y}^{ \overline{z} } \left( (1-k) y + k t - c \right) f^{S}_{(m-1), (m)} (y, t ) \mathrm{d}t \end{aligned}$$

which simplifies to:

$$\begin{aligned} \pi '_{S}( y )&= - \left( y - c \right) f^{S}_{(m)} (y) + (1-k) \int \limits _{y}^{\overline{z}} \int \limits _{\underline{z}}^{y} f^{S}_{(m-1),(m) } (s,t) \mathrm{d}s \mathrm{d}t\nonumber \\&= -\left( y - c \right) f^{S}_{(m)}(y) + (1-k) N (y) \end{aligned}$$

(29)

In (29) $f^{S}_{(m)}(y)$ is the density of $\zeta _{(m)}$ evaluated at an ask $a=y$:

$$\begin{aligned} f^{S}_{(m)}(y) = m H(y) g_{b}(y) + (n-1) Q(y) g_{s}(y) \end{aligned}$$

(30)

where:

$$\begin{aligned} H(y) \!&= \!\! \sum _{\begin{array}{c} i + j = m \!-\!1 \\ 0 \le i \le m\!-\!1 \\ 0 \le j \le n-1 \end{array} } \left( {\begin{array}{c}m\!-\!1\\ i\end{array}}\right) \left( {\begin{array}{c}n\!-\!1\\ j\end{array}}\right) G_{b}(y)^{i} G_{s}(y)^{j} ( 1\! \!-\!\! G_{b} (y) ) ^{m-1-i} ( 1 \!-\! G_{s} (y) ) ^{n-1-j}\\ Q(y) \!&= \!\! \sum _{\begin{array}{c} i + j = m -1 \\ 0 \le i \le m \\ 0 \le j \le n\!-\!2 \end{array} } \left( {\begin{array}{c}m\\ i\end{array}}\right) \left( {\begin{array}{c}n\!-\!2\\ j\end{array}}\right) G_{b}(y)^{i} G_{s}(y)^{j} ( 1 \!-\! G_{b} (y) ) ^{m-i} ( 1 \!-\! G_{s} (y) ) ^{n\!-\!2-j} \end{aligned}$$

In (29) $N (y)$ corresponds to:

$$\begin{aligned} N (y)&= \int \limits _{y}^{\overline{z}} \int \limits _{\underline{z}}^{y} f^{S}_{(m-1),(m) } (s,t) \mathrm{d}s \mathrm{d}t = P(\zeta _{(m-1)} < a=y < \zeta _{(m)})\\&= \sum _{\begin{array}{c} i + j = m-1 \\ 0 \le i \le m \\ 0 \le j \le n-1 \end{array} } \left( {\begin{array}{c}m\\ i\end{array}}\right) \left( {\begin{array}{c}n-1\\ j\end{array}}\right) G_{b}(y)^{i} G_{s}(y)^{j} ( 1 - G_{b} (y) ) ^{m-i} ( 1 - G_{s} (y) ) ^{n-1-j} \end{aligned}$$

Therefore:

$$\begin{aligned} \pi '_{S}( y ) = -\left( y - c \right) \left( m H(y) g_{b}(y) + (n-1) Q(y) g_{s}(y) \right) + (1-k) N(y) \end{aligned}$$

(31)

Functions $f^{B}_{(m)}(y)$, $f^{S}_{(m)}(y)$, $K(y)$, $L(y)$, $M(y)$, $H(y)$, $Q(y)$, and $N (y)$ are functions of $G_{b}$ and $G_{s}$, the cumulative distribution function of other buyers’ bids and sellers’ asks, respectively, and $g_{b}$ and $g_{s}$, the corresponding density functions. They can be computed using the formulas for the density of order statistics (David Nagaraja 2003).

Plugging (28) in (24) and (31) in (25), we obtain:

$$\begin{aligned}&q'_{B} ( y ) = \lambda \left\{ \left( v - y \right) \left( n K(y) g_{s}(y) + (m-1) L(y) g_{b}(y) \right) - k M(y) \right\} q_{B}(y ) \end{aligned}$$

(32)

$$\begin{aligned}&q'_{S} ( y ) = \lambda \left\{ -\left( y - c \right) \left( m H(y) g_{b}(y) + (n-1) Q(y) g_{s}(y) \right) + (1-k) N(y) \right\} q_{S}(y ) \nonumber \\ \end{aligned}$$

(33)