Eliciting beliefs in continuous-choice games: a double auction experiment

Abstract

This paper proposes a methodology to implement probabilistic belief elicitation in continuous-choice games. Representing subjective probabilistic beliefs about a continuous variable as a continuous subjective probability distribution, the methodology involves eliciting partial information about the subjective distribution and fitting a parametric distribution on the elicited data. As an illustration, the methodology is applied to a double auction experiment, where traders’ beliefs about the bidding choices of other market participants are elicited. Elicited subjective beliefs are found to differ from proxies such as Bayesian Nash equilibrium beliefs and empirical beliefs, both in terms of the forecasts of other traders’ bidding choices and in terms of the best-response bidding choices prescribed by beliefs. Elicited subjective beliefs help explain observed bidding choices better than BNE beliefs and empirical beliefs. By extending probabilistic belief elicitation beyond discrete-choice games to continuous-choice games, the proposed methodology enables to investigate the role of beliefs in a wider range of applications.

Appendices

Appendix 1: Figures and Tables

Fig. 1

BNE beliefs: bid and offer density functions. According to the BNE strategies, the probability of trading is zero for buyers with a very low value and sellers with a very high cost. Therefore, their misrepresentation of private values and costs cannot be bounded. In a market with four buyers and four sellers, buyers with value lower than $1.05 and sellers with cost higher than $8.95 can never trade. In a market with three buyers and three sellers, buyers with value lower than $1.31 and sellers with cost higher than $8.69 can never trade. In a market with two buyers and two sellers, buyers with value lower than $1.72 and sellers with cost higher than $8.28 can never trade

Table 7 Sample distribution (in percent) of the subjective probabilities assigned to each interval

Table 8 Average absolute difference between subjective and BNE beliefs

Table 9 Average absolute difference between subjective and empirical beliefs

Table 10 Average absolute difference between subjective beliefs and BNE beliefs

Table 11 Average absolute difference between subjective beliefs and empirical beliefs

Table 12 Absolute difference (in dollars) between the expected payoff generated by the best response to subjective beliefs and the best response to BNE beliefs (column 1–3) and absolute difference between the expected payoff generated by the best response to subjective beliefs and the best response to empirical beliefs (column 4–6)

Table 13 Percentage of observations for which choice is consistent with the best response to subjective beliefs, empirical beliefs, or BNE beliefs

Table 14 Accuracy of beliefs. Medians in round 1 and round 15. Comparison of subjective beliefs, BNE beliefs, and empirical beliefs according to different measures of accuracy: quadratic score, linear score and average absolute difference ($AAD$)

Table 15 Calibration: empirical relative frequency of bids (offers) falling in each interval, by subjective beliefs. Subjective beliefs are represented by the vector of probabilities assigned to each interval

Appendix 2: The double auction model

This section provides an overview of the double auction model by Rustichini et al. (1994).

The market is populated with $n$ buyers and $n$ sellers. Each buyer has an endowment of $10 and each seller has one unit of an indivisible good. Each buyer can purchase at most a single unit of the good from any seller, and each seller could sell a single unit of the good to any buyer. Each buyer has a private value and each seller has a private cost for the good. Private values and costs are independently drawn from the uniform distribution over $[0,10]$. A subject’s private value or cost is her own private information, and the process by which private values and costs are drawn is common knowledge among subjects.

All buyers and sellers simultaneously submit their bids or offers. Sorting all bids and offers in increasing order as $\psi _{(1)} \le \psi _{(2)} \le \cdots \le \psi _{(2n)}$, the market price is set at the midpoint between the middle two figures, at $p = 0.5 \psi _{(n)} + 0.5 \psi _{(n+1)}$.⁶¹ Buyers who submit a bid larger than or equal to $p$ and sellers who submit an offer smaller than or equal to $p$ trade. If trading at $p$, a buyer with private value $v$ earns a payoff equal to $v-p$, and a seller with private cost $c$ earns a payoff equal to $p-c$. Participants who do not trade earn zero.⁶²

Fig. 2

Example: a market with three buyers and three sellers. Note: Buyers bid at 9.50, 5.00 and 4.20 and sellers offer at 1.00, 3.50 and 5.30. Bids and offers imply a demand and supply curve, respectively. The crossing of the demand and supply curves determines the interval [4.20, 5.00], from which a market-clearing price can be selected. The market-clearing price is selected at the midpoint of the interval, $p= 4.60$

Lets consider a buyer $i$ with private value $v$. According to the auction rules, the relationship between her bid $b$ and the bids and offers of the other $2n-1$ participants determines what her payoff will be, depending on whether she trades and, if so, at what price. Sorting all other $2n-1$ bids and offers in increasing order as $\zeta _{(1)} \le \zeta _{(2)} \le \cdots \le \zeta _{(2n-1)}$, buyer $i$ trades if $b > \zeta _{(n)}$. If $\zeta _{(n)} < b < \zeta _{(n+1)}$, the price is $p=0.5\zeta _{(n)}+ 0.5 b$. If $b > \zeta _{(n+1)}$, the price is $p=0.5\zeta _{(n)}+ 0.5 \zeta _{(n+1)}$. Buyer $i$ chooses bid $b\ge 0$ without knowing the other 2$n$-1 bids and offers but holding probabilistic beliefs about them.⁶³

Assuming buyer $i$ behaves as a risk-neutral subjective expected utility maximizer, given her beliefs about the realizations of $\zeta _{(n)}$ and $\zeta _{(n+1)}$, represented by the joint density $f_{(n),(n+1)}$, she chooses bid $b$ in order to maximize her expected payoff:

$$\begin{aligned} \pi (v,b)&= \int _{b}^{10} \int _{0}^{b} \left( v - [0.5s+ 0.5 b ] \right) f_{(n),(n+1)} (s,t) ds dt \\ &\quad + \int _{0}^{b} \int _{0}^{t} \left( v - [0.5s+ 0.5 t ] \right) f_{(n),(n+1)} (s,t) ds dt , \end{aligned}$$

(1)

where $\zeta _{(n)}$ and $\zeta _{(n+1)}$ are defined over the set $[0,10]$, the first term is the expected payoff if $\zeta _{(n)} < b < \zeta _{(n+1)}$ and the second term is the expected payoff if $b > \zeta _{(n+1)}$. The inner integral integrates over all possible values of $\zeta _{(n)}$ and the outer one over all possible values of $\zeta _{(n+1)}$.

Analogously, the beliefs of seller $i$, who has private cost $c$ and submits offer $a$, can be represented by $f_{(n-1),(n)}$, the joint density of $\zeta _{(n-1)}$ and $\zeta _{(n)}$, and her expected payoff can be written as

$$\begin{aligned} \pi (c,a)&= \int _{a}^{10} \int _{0}^{a} \left( 0.5 a + 0.5 t - c \right) f_{(n-1),(n)}(s,t) ds dt\\&\quad + \int _{s}^{10} \int _{a}^{10} \left( 0.5s + 0.5 t - c \right) f_{(n-1),(n)}(s,t) ds dt . \end{aligned}$$

(2)

where $\zeta _{(n-1)}$ and $\zeta _{(n)}$ are defined over the set $[0,10]$, the first term is the expected payoff if $\zeta _{(n-1)} < a < \zeta _{(n)}$ and the second term is the expected payoff if $a < \zeta _{(n-1)}$. The inner integral integrates over all possible values of $\zeta _{(n-1)}$ and the outer one over all possible values of $\zeta _{(n)}$.

Summing up, a buyer’s beliefs are represented by joint density $f_{(n),(n+1)}$ and a seller’s beliefs are represented by joint density $f_{(n-1),(n)}$. Denoting the density functions of other buyers’ bids and other sellers’ offers with $g_{b}$ and $g_{s}$ respectively, and their cumulative density functions with $G_{b}$ and $G_{s}$ respectively, $f_{(n),(n+1)}$ and $f_{(n-1),(n)}$ can be written as function of the beliefs about other buyers’ bids ($g_b$ and $G_b$) and the beliefs about other sellers’ offers ($g_s$ and $G_s$).

To show it, consider a general formulation with $m$ buyers and $n$ sellers. Denote $k=m$. Denote with $f_{(k),(k+1)}$ the joint density of the $m$th and ($m+1$)th order statistics of the $m+n-1$ bids and offers. Then:

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

Denote $k=m-1$. Denote with $f_{(k),(k+1)}$ the joint density of the ($m-1$)th and $m$th order statistics of the $m$+$n$-1 bids and offers. Then:

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

The formulas for the marginal densities of order statistics are as follows. Denote $k=m$.

For the buyer:

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

For the seller:

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

Appendix 3: Parametric fitting of beliefs

Depending on (a) the number of intervals where positive probability is placed, and (b) whether the intervals, where positive probability is placed, are adjacent to each other, I fit:

• (1) a triangular distribution when positive probability is placed (i) over a single interval or (ii) over two adjacent intervals,

• (2) the union of two triangular distributions when positive probability is placed (i) over two non-adjacent intervals or (ii) over three intervals of which two but not all three are adjacent (iii) over four intervals, consisting of two disjoint pairs of adjacent intervals,

• (3) the union of three triangular distributions when positive probability is placed over three interval none of which is adjacent to any other,

• (4) a unimodal beta distribution when positive probability is placed over three or more intervals all adjacent to each other,

• (5) the union of a beta distribution and a triangular distribution when positive probability is placed over more than three intervals, of which at least three, but not all, are adjacent to each other.

Here below I describe how the fitting is performed in each case. Figure 3 shows an example for each case, the most common being the case in which beliefs are fitted with a unimodal beta distribution (approximately 90 % of observations), followed by the case in which beliefs are fitted with a triangular distribution (approximately 7 % of observations).

1.1 Triangular distribution

If positive probability is assigned to only one interval, $[l,r]$, then I assume that the support of the subjective distribution is $[l,r]$ and I fit a triangular distribution over it. The fitted isosceles triangle has base $r-l$ and height $\frac{2}{r-l}$. If positive probability is assigned to two adjacent intervals and if equal probability is assigned to each interval, then I assume that the support of the subjective distribution is the union of the two intervals and the fitted isosceles triangle has base $4$ and height $\frac{1}{2}$.

If positive probability is assigned to two adjacent intervals and a higher probability is assigned to one interval than to the other, then I assume that the subjective distribution has the shape of an isosceles triangle and that its support contains entirely the interval that was assigned a higher probability and partly the other interval. If the subject assigns probability $\alpha$ and $1-\alpha$ to the intervals $[y, y+2]$ and $(y+2, y+4]$, respectively, where $\alpha <0.5$, then, the fitted isosceles triangle has base with endpoints $y+2-t$ and $y+4$ and height $h=\frac{2}{t+2}$, with $t = \frac{2 \sqrt{\frac{\alpha }{2}} }{1 - \sqrt{\frac{\alpha }{2}} }$.⁶⁴

1.2 Union of two triangular distributions

If positive probability is assigned to two non-adjacent intervals, then I assume that the support of the subjective distribution is the union of the two intervals and I fit a triangular distribution over each interval. For example, probability $\alpha$ is assigned to the interval $[l_1,r_1]$ and probability $1-\alpha$ to the interval $[l_2,r_2]$, then I assume that the support of the distribution is the union of $[l_1,r_1]$ and $[l_2,r_2]$. The isosceles triangle fitted over $[l_1,r_1]$ has base $r_1-l_1$ and height $\frac{2 \alpha }{r_1-l_1}$. The isosceles triangle fitted over $[l_2,r_3]$ has base $r_2-l_2$ and height $\frac{2 (1-\alpha )}{r_2-l_2}$.

If positive probability is assigned to three intervals of which two but not all three are adjacent, then I assume that the support of the subjective distribution is the union of the intervals and I fit two triangular distributions, one over the two adjacent intervals and another other the non-adjacent interval. For example, suppose that the intervals $[l_1,r_1]$, $[l_2,r_2]$ and $[l_3,r_3]$ are assigned probability $\alpha$, $\beta$ and $1 - \alpha - \beta$, respectively, and that $[l_1,r_1]$ and $[l_2,r_2]$ are adjacent to each other (with $l_2>l_1$), while $[l_3,r_3]$ is not adjacent to any of the other intervals. Then one triangle is fitted over the union of $[l_1,r_1]$ are $[l_2,r_2]$ and one triangle is fitted over $[l_3,r_3]$, following the procedures already described for fitting one triangular distribution.⁶⁵

If positive probability is assigned to four intervals, consisting of two disjoint pairs of adjacent intervals, then I also fit two triangular distributions. Each triangular distribution has support over a pair of adjacent intervals and the fitting is done following the procedure already described for fitting one triangular distribution.

1.3 Union of three triangular distributions

If positive probability is assigned to three intervals none of which is adjacent to any other, then I assume that the support of the subjective distribution is the union of the intervals and I fit a triangular distribution over each intervals, following the procedure already described for fitting one triangular distribution.

1.4 Unimodal beta distribution

If positive probability is assigned to three or more intervals all adjacent to each other, then I fit a generalized unimodal Beta distribution over the intervals. The cumulative distribution function for a unimodal Beta distribution evaluated at $x$ is denoted $Beta(x,\alpha ,\beta ,l,r)$, where $\alpha$ and $\beta$ are shape parameters and $l$ and $r$ are location parameters determining the support for the distribution over the range $[l,r]$. If a subject does not assign positive probability to the right tail interval $[10.01,\infty )$, then the lower bound $l$ of the support for the fitted Beta distribution will coincide with the left endpoint of the leftmost interval with positive probability and the upper bound $r$ of the support will coincide with the right endpoint of the rightmost interval with positive probability and, therefore, the parameters $l$ and $r$ will be fixed. Thus, fitting the data with a Beta distribution requires solving the problem $\min _{\begin{array}{c} \alpha ,\beta \end{array}} \sum _{j=1}^{6} [Beta(r_{j},\alpha ,\beta ,l,r) - G(r_{j}) ]^2$, where $G(r_{j})$ is the sum of the subjective probabilities assigned up to the interval with right endpoint $r_j$, inclusive.

If instead a subject assigns positive probability to the upper unbounded interval [$10.01,$\infty$), I let the location parameter $r$ be a free parameter in the minimization of the least squares problem. I restrict $r$ to lie within the most extreme value recorded in the data, which is 18.50. Thus, the problem becomes $\min _{\begin{array}{c} \alpha ,\beta ,r<18.50 \end{array}} \sum _{j=1}^{6} [Beta(r_{j},\alpha ,\beta ,l,r) - G(r_{j}) ]^2$.

1.5 Union of a Beta distribution and a triangular distributions

If positive probability is assigned to more than three intervals, of which at least three but not all are adjacent to each other, then I fit a unimodal beta distribution over the three or more adjacent intervals and a triangular distribution over the remaining one or two intervals. I follow the procedures already described for fitting a triangular distribution and a unimodal Beta distribution.

Fig. 3

A selection of the fitting methods 1–5

Goodness of fit is assessed by the average absolute deviation between the fitted and the elicited beliefs. In most cases the average absolute deviation is below 0.01, and in all cases below 0.06.

Appendix 4: Descriptive Results

1.1 Descriptive results about bidding behavior

In this section I illustrate how the observed bidding choices compare with the bidding behavior prescribed by the risk-neutral BNE model, as presented by Rustichini et al. (1994).⁶⁶ Figure 4 illustrates the approximate risk-neutral bidding functions predicted by the model.

Fig. 4

Approximate risk-neutral BNE bid and offer functions

Since the double auction is a strategic environment with private information, strategic misrepresentation of private information is a key feature within the BNE model. As a measure of how much subjects reveal of the private information they hold, I use the value underrevelation ratio and the cost underrevelation ratio, as defined by Cason and Friedman (1997). The value underrevelation ratio $VUR(v,b)$ is the fraction of the buyer’s private value, $v$, discounted in the chosen bid, $b$, and the cost underrevelation ratio $CUR(c,a)$ is the cost mark-up relative to the highest possible cost. Thus, $VUR(v,b)= (v-b)/v$ and $CUR(c,a)= (a-c)/(9.99-c)$.⁶⁷ A positive ratio corresponds to underrevelation and a negative ratio corresponds to overrevelation.⁶⁸

Table 16 Underrevelation of private information and deviation of the BNE best response from the observed choice

The upper panel of Table 16 reports the median value and cost underrevelation ratios. Results are reported for (i) the entire sample, and separately for (ii) the subsample in which underrevelation occurs and (iii) the subsample in which overrevelation occurs. Within subsample (ii) the median $VUR(v,b)$ and $CUR(c,a)$ are 7 and 4 %, respectively.⁶⁹

The lower panel of Table 16 reports the magnitude of the percent deviation of the BNE best response from the observed choice, defined as $D(b_{BNE},b)=(b_{BNE}-b)/b_{BNE}$ for buyers and $D(a_{BNE},a)=(a_{BNE}-a)/a_{BNE}$ for sellers. Within subsample (ii), the median deviation is approximately 0 for buyers and 2 % for sellers. ⁷⁰

1.2 Descriptive results about market performance

Table 17 reports a description of market performance across the experimental auctions. The number of trades taking place in each round ranges between zero and three, with one or two trades per round being the most common outcome. Over all observations, including those when no trade occurs, mean profits are $0.88. In a competitive equilibrium (CE) buyers and sellers submit respectively their private values and private costs without engaging in any strategic misrepresentation of their private information. Across all experimental auctions, trading efficiency, defined as the percentage of the gains from exchange realized by traders in comparison with the CE, is 78 % and prices are within the CE price interval in 50 % of the rounds.⁷¹

Table 17 Market performance