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.

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Notes

  1. 1.

    In particular, a good fit is often achieved when the parameters of the distribution of payoff perturbations are estimated in order for the predicted outcome distribution to match the data as well as possible.

  2. 2.

    In a \(k\)-double auction with \(m\) buyers and \(n\) sellers (\(n,m \ge 2\)) the trading price is set at \(p = (1-k) \psi _{(m)} + k \psi _{(m+1)}\) with \(k \in [0,1]\) and bids and asks ordered in a list as \(\psi _{(1)} \le \psi _{(2)} \le ... \le \psi _{(m+n)} \). Buyers with a bid larger than or equal to \(p\) and sellers with an ask smaller than or equal to \(p\) will trade. In particular, if \(k = 1\) (\(k = 0\)), the double auction is called a buyer’s-bid auction (seller’s-ask auction) because in the bilateral case (\(n=m=1\)) the buyer’s bid (the seller’s ask) is the price whenever trade occurs.

  3. 3.

    If excess demand or excess supply arises, priority is given to sellers whose asks are smallest and to buyers whose bids are largest. A fair lottery then determines who trades among the remaining subjects on the long side of the market.

  4. 4.

    \(y_L=0\) if conditions (19)–(23) hold on \((0,y_0]\).

  5. 5.

    \(y_U=1\) if conditions (18)–(20), (22), (23) hold on \([y_0,1)\).

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Correspondence to Claudia Neri.

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)

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Neri, C. Quantal response equilibrium in a double auction. Econ Theory Bull 3, 79–90 (2015). https://doi.org/10.1007/s40505-014-0038-4

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Keywords

  • Quantal response equilibrium
  • Existence of equilibrium
  • Auctions

JEL Classification 

  • C62
  • C72
  • D44
  • D82