Randomized double auctions: gains from trade, trader roles, and price discovery

Abstract

Experimental double-auction commodity markets are known to exhibit robust convergence to competitive equilibria under stable or cyclical supply and demand conditions, but little is known about their performance in truly random environments. We provide a comprehensive study of double auctions in a stochastic setting where the equilibrium prices, trading volumes and gains from trade are highly variable across periods, and with commodity traders who may buy or sell their goods depending on market conditions and their individual outcomes. We find that performance in this stochastic environment is sensitive to underlying market conditions. Efficiency is higher and convergence to the competitive equilibrium stronger when the potential gains from trade are high and when the equilibrium spans a wide range of quantities, implying a large number of marginal trades. Speculative re-trading is prevalent, especially among those who have little to gain under equilibrium pricing. Those with the largest expected gains typically earn far less than predicted, while those with little or no predicted earnings gain modestly from speculation, leading to some redistribution of gains from high to low expected earners. Excessive trading volumes are associated with negative efficiencies in markets with low gains from trade, but not in the high-gains markets, where zero-sum trading and re-trading appear to enforce efficiency and near-equilibrium pricing. Buyers earn more relative to their competitive equilibrium benchmark than sellers do. Introducing trader specialization leads to fewer trading errors and higher market efficiency, but it does not eliminate zero-sum trading and re-trading.

Appendices

Appendix A: equilibrium predictions

Table 13 Equilibrium price predictions and empirical frequency by average yield realization

Equilibrium prices Since all traders have the same valuation schedules, the equilibrium price $P^{eq}$ in each period equates mean supply and mean demand, and is determined by the marginal unit value, as given in Table 1, at the average yield $\bar{Y_t} = \frac{1}{N}\sum _i Y_{it}$. The mapping from average market yields to equilibrium prices is displayed in Table 13. The equilibrium price spans a range of values when the average yield is an integer between 1 and 6; in this case market supply and demand curves overlap vertically, see, e.g., Fig. 1, market periods 1, 4, 19 and 20. The equilibrium price takes a single value for average yields that are non-integer or above 6 units; in this case market supply and demand curves overlap horizontally, yielding a range of equilibrium trading volumes; see Fig. 1, other market periods.

Equilibrium trading volumes The equilibrium trading volumes are determined by the intersection of the net market supply and demand curves. If the average market yield is an integer between 1 and 6 units, the market supply and demand curves intersect over a vertical interval, as discussed above, yielding a unique prediction for the equilibrium trading volume:

$$\begin{aligned} Q^{eq}=\sum _{{i: Y_{it}>\bar{Y_t}}} (Y_{it}-\bar{Y_t})= \sum _{j: Y_{jt}<\bar{Y_t}} (\bar{Y_t}-Y_{jt}), \end{aligned}$$

(2)

where $\{i: Y_{it}>\bar{Y_t}\}$ is the set of net sellers, and $\sum _{{i: Y_{it}>\bar{Y_t}}} (Y_{it}-\bar{Y_t})$ is the net market supply; $\{j: Y_{jt}<\bar{Y_t}\}$ is the set of net buyers, and $\sum _{j: Y_{jt}<\bar{Y_t}} (\bar{Y_t}-Y_{jt})$ is the market demand, as predicted by the equilibrium theory.

If the average market yield is not an integer, or is greater than 6 units, the net market supply and demand overlap over a horizontal range, yielding a range of volumes $[Q_L,Q_U]$ consistent with the equilibrium prediction. This happens because of the discrete nature of unit values: while $Q_L$ is the lowest number of trades necessary to fully realize all potential gains from trade, additional marginal trades may be available in equilibrium; traders may be holding units that they are just indifferent between trading and not trading, as any such trade would bring zero profit.

The lower and upper bounds of equilibrium trading volumes, $Q_L$ and $Q_U$, are given by:

$$\begin{aligned} Q_L= & {} \max \{ \min Q^S, \min Q^D \} , \end{aligned}$$

(3)

$$\begin{aligned} Q_U= & {} \min \{ \max Q^S, \max Q^D \}. \end{aligned}$$

(4)

In the above, the bounds on market supply and demand quantities $Q^S$ and $Q^D$, evaluated at the equilibrium price, are calculated as follows. If $\bar{Y_t}<6$ then $P^{eq}>0$. Hence:

$$\begin{aligned} \min Q^S= & {} \sum _{{i: Y_{it}>\bar{Y_t}}} (Y_{it}-\lceil \bar{Y_t}\rceil ), {\;\;\;} \max Q^S =\sum _{{i: Y_{it}>\bar{Y_t}}} (Y_{it}-\lfloor \bar{Y_t}\rfloor ), \end{aligned}$$

(5)

$$\begin{aligned} \min Q^D= & {} \sum _{j: Y_{jt}<\bar{Y_t}} (\lfloor \bar{Y_t}\rfloor -Y_{jt}), {\;\;\;} \max Q^D = \sum _{j: Y_{jt}<\bar{Y_t}} (\lceil \bar{Y_t}\rceil -Y_{jt}), \end{aligned}$$

(6)

where $\lfloor \bar{Y_t}\rfloor \equiv$ floor$(\bar{Y_t})$ and $\lceil \bar{Y_t}\rceil \equiv$ ceil$(\bar{Y_t})$ denote the floor and the ceiling of $\bar{Y_t}$, respectively.

If $\bar{Y_t}>6$ then $P^{eq}=0$, and the market supply $Q^S$ at zero price is bounded from below by zero, while the market demand $Q^D$ is unbounded from above; and the upper bound of supply and the lower bound of demand are determined by the condition that each trader should hold at least 6 units in equilibrium. Hence:

$$\begin{aligned}&\min Q^S =0, {\;\;\;} \max Q^S =\sum _{{i: Y_{it}>6}} (Y_{it}-6), \end{aligned}$$

(7)

$$\begin{aligned}&\min Q^D = \sum _{j: Y_{jt}<6} (6-Y_{jt}), {\;\;\;} \max Q^D = \infty . \end{aligned}$$

(8)

Example 1

Consider Session 1, Period 1 individual yields for 12 traders as presented in Table 14. $\bar{Y_t}=4$, and each trader will hold exactly 4 units in equilibrium; their opportunity cost of selling the fourth unit is 125, and their maximal willingness to pay for the fifth unit is 75 (Table 1), giving rise to the equilibrium price range of [75, 125] at which no strictly profitable trades are possible. The equilibrium trading volume is uniquely determined at 13 units; see Fig. 1, Period 1 for illustration.

Example 2

Now consider Session 1, Period 5 individual yields (Table 14). $\bar{Y_t}=8$, hence in equilibrium $P^{eq}=0$, all traders hold at least 6 units, and no strictly profitable trades are available for any trader. The equilibrium trading volume ranges between 2 and 26 units, depending on whether the traders are willing or not to engage in marginal zero-profit trades. See Fig. 1, Period 5.

Example 3

Consider further individual yields in Session 1, Period 7 (Table 14). $\bar{Y_t}=2.25$, with nine traders holding two units, and three traders holding three units in equilibrium. Traders holding two units are willing to buy the third unit at a price no more than their marginal value, 175; traders holding three units are willing to sell their third unit at a price no less than their opportunity cost, 175. Hence the equilibrium price is 175, allowing no strictly profitable trades at an equilibrium allocation. The equilibrium trading volume ranges between 7 and 10 units, depending on whether the traders are willing or not to engage in marginal zero-profit trades. See Figure 1, Period 7.

We emphasize again that if net market supply and demand overlap horizontally, as in Examples 2 and 3 above, then the equilibrium allocation of units among traders is not unique, and marginal zero-gain trades between traders are still possible in equilibrium, as long as such trades lead to another allocation consistent with equilibrium.

Table 14 Illustration: market yields and equilibrium predictions

Appendix B: net trade and re-trade earnings calculation

The number of each trader’s net trade and re-trade transactions in a given period, and her period earnings from net trade and re-trade, are calculated as follows. Let be $n_b$ and $n_s$ the number of trader’s buy and sell transactions, and let $n_{all}= n_b+n_s$ be her total number of transactions. Then the number of net trades is:

$$\begin{aligned} n_{net}=\max \{ n_b, n_s\} - \min \{ n_b, n_s\}, \end{aligned}$$

(9)

and the number of re-trades is:

$$\begin{aligned} n_r=n_{all}-n_{net}= 2\times \min \{ n_b, n_s\}. \end{aligned}$$

(10)

Further, let $\Pi ^{all}$ be this trader’s total earnings in a period, and let ${\bar{p}}_b$ and ${\bar{p}}_s$ be her average buying and selling prices. Then her earning from re-trade are calculated as

$$\begin{aligned} \Pi ^{r} = (n_r/2) \times ({\bar{p}}_s -{\bar{p}}_b)= \min \{ n_b, n_s\} \times ({\bar{p}}_s -{\bar{p}}_b). \end{aligned}$$

(11)

The remaining trader earnings are attributed to net trades:

$$\begin{aligned} \Pi ^{net}=\Pi ^{all}-\Pi ^{r}. \end{aligned}$$

(12)

Equivalently, the earnings from net trade can be calculated in the following way. Let i be the initial number of units on hand, and let k be the number on hand at the end of trading. Then $k>i$ with $n_{net}=k-i$ for net buyers; $i>k$ with $n_{net}=i-k$, for net sellers; and $i=k$ for net no-traders. Let $v_j$ denote j-th unit value, as presented in Table 1 in the main paper.

The profit from net trade for net buyers is then calculated as:

$$\begin{aligned} \Pi ^{net}=\sum _{j=i+1}^{k} v_j -n_{net}\times {\bar{p}}_b, \end{aligned}$$

(13)

and the profit from net trade for net sellers is calculated as:

$$\begin{aligned} \Pi ^{net}=n_{net}\times {\bar{p}}_s-\sum _{j=k+1}^{i} v_j. \end{aligned}$$

(14)

The profit from re-trade is then calculated as the difference between total profit and net profit, $\Pi ^{r}=\Pi ^{all}-\Pi ^{net}$, or, equivalently, using expression 11 above.

For example, suppose a trader initially held $i=2$ units in a period. She sold two units, $n_s=2$, at the prices of 130 and 180, and bought three units, $n_b=3$, at the prices of 150, 100 and 80. This trader is a net buyer with one net trade, $n_{net}=3-2=1$, and four re-trades, $n_r=(2+3)-1=4$, or two re-trade pairs: $\min \{ n_b, n_s\}=2$, holding $k=3$ units at the end of trading. The average selling price is ${\bar{p}}_s = 155$, and the average buying price is ${\bar{p}}_b= 110$. Trader total earnings in this period are $\Pi ^{all}=105$, with earnings from net trade $\Pi ^{net}=v_3-1*(110)=125-110=15$, and earnings from re-trade $\Pi ^{r}=2*(155-110)=90$.