Economic analysis of the double auction

We assume that each agent i has a CES (Constant Elasticity of Substitution) utility function,  

$$U(x)=\left( \sum _{g=1}^m \alpha_g x_g^\rho \right)^{1 \over \rho},$$ (4)

where $x=(x_1, \ldots, x_m)$ is a vector of goods, the are preference weights, and is the substitution parameter. We choose the CES functional form for its convenience and generality--including quadratic, logarithmic, linear, and many other forms as special cases.

In constructing agent strategies, we dictate that they always choose actions leading to nonnegative payoffs. We can characterize this in terms of the agents' reservation prices [25]. The reservation price is defined as the maximum (minimum) price an agent is willing to pay (accept) for the good it wants to buy (sell). We can define agent i's buying and selling reservation prices, $\bar{P}_{b}$ and $\bar{P}_{s}$, as the prices such that its utility keeps constant when buying or selling one unit of good.

It can be easily shown that for quasi-concave utility (such as CES), the agent's reservation buy price is always lower than its reservation sell price.

Theorem 1721

 For any agent with strictly quasi-concave utility function, the agent's reservation buy price $\bar{P}_{b}$ is always less or equal to its reservation sell price $\bar{P}_{s}$.

Proof: Let an agent's utility function be U(x_j, x_m, x_-j,-m) where j is the good for current auction, m is the numerair good to be traded with good j, x_-j,-m is agent's holding of goods other than j and m.

Assuming $U(x_j, x_m, x_{-j,-m})=\bar{U}$, by definition of reservation price, we have $U(x_j-1, x_m+\bar{P}_{s},x_{-j,-m})=\bar{U}$ and $U(x_j+1, x_m-\bar{P}_{b},x_{-j,-m})=\bar{U}$. Since x_-j,-m and $\bar{U}$ are constants, we can re-write the previous equations to express x_m as a function of x_j:

Adding  (7) and (8) we have $\bar{P}_{s}-\bar{P}_{b}=f(x_j+1)+f(x_j-1)-2x_m$.

Since the utility function U(x_j, x_m, x_-j,-m) is quasi-concave,
$U(x_j, x_m, x_{-j,-m})=\bar{U}$ yields a convex function x_m=f(x_j). By definition of convex function, $f(x_j+1)+f(x_j-1)\geq 2f(x_j)$. Therefore $\bar{P}_{s}\geq \bar{P}_{b}$. $\Box$

Since the utility function is increasing in each of its components, for any trading price P such that $P \geq \bar{P}_{s}$ or $P \leq \bar{P}_{b}$, conducting the trade can never degrade the agent's utility. Therefore, if an agent submits bids such that its utility never decrease, its sell bid P_b and buy bid P_s should satisfy $P_s \geq \bar{P}_{s}$and $P_b \leq \bar{P}_{b}$. It follows then from Theorem 1 that we always have $P_b\leq P_s$. From this we can further show that an agent will never be both a buyer or seller in the same market.

  

Figure 2: An agent's reservation prices and its actual bids

Theorem 1750

Given that an agent's sell bid exceeds its buy bid, the agent can never be matched both as buyer and a seller on the same market.

 Proof: Let the agent be i. Suppose agent i can be matched to agent j as a buyer, and be matched to agent k as a seller. Then we have $P_{b}^{i}\geq P_{s}^{j}$ and $P_{b}^{k}\geq P_{s}^{i}$. Since agent i's bids satisfy $P_b^i\leq P_s^i$,we have  

$$P_{s}^{j} \leq P_{b}^{i} < P_{s}^{i} \leq P_{b}^{k}.$$ (5)

According to the matching rule in our double auction, the highest buyer is matched to the lowest seller, the 2nd highest buyer is matched to the 2nd lowest seller and so on. Thus for any 2 pairs of traders, if the buy price in the first pair is higher than the buy price in the second pair, the sell price in the first pair is always lower than the one in the second pair. Now we have two pairs of traders {buyer k, seller i}, {buyer i, seller j}. If $P_{b}^{k} \geq P_{b}^{i}$, then we will have $P_{s}^{i}\leq P_{s}^{j}$. This contradicts the inequality (9). $\Box$