Convergence

By convergence, we mean that the system reaches a termination condition. In our system, termination is mandated when, for all goods, all the buy prices are lower than all the sell prices. That means no agents can be matched as a buyer or a seller, and no trading can happen.

The trading process in our auction can be cast as a Edgeworth process in the broad sense that agents' utilities continually increase throughout the process [25]. However, the specific trading rules in our auction differ in a significant way from the ones in standard Edgeworth process [24]. In a standard Edgeworth process, a central price is announced at each period by the auctioneer, and each agent decides whether to trade based on that price. Thus agents have no role in determining the prices in the market. This leaves little room for strategic behavior. Uzawa [24] proved that such a process converges to a Pareto optimum distribution where no price exists such that at least two agents can increase their utilities by trading.

In our double auction, agents submit bidding prices which will determine the trading prices. Therefore agents have direct influence on the market, and they have incentives to act strategically. Such behavior may cause the auction end prematurely before reaching a Pareto optimum. To see how this can happen, consider an auction that is one step from the Pareto optimum, with the only trading opportunity between buyer A and seller B left. Now suppose A's reservation buy price is 5, B's reservation sell price is 4. If A behaves strategically, A may offer a price far lower than 5, say at 3. Then the market will terminate and agent A and B will never get a chance to trade.

Below we prove that the synchronous double auction market converges. The proof relies on the simple fact that all agents' utilities keeps increasing in the auction.

Theorem 1826

The synchronous double auction market converges.

Proof: Let $U_t=\sum_{i=1}^nU^i(e_t^i)$be the sum of utilities of all agents. Let $\bar{U}=\sum_{i=1}^nU^i(E)$, where E is the total endowment in the system. In a pure exchange system such as our auction market, the total endowment E remains constant. For each agent i, $e_t^i\leq E$. Thus $U_t \leq \bar{U}$ for any t, that is, U_t is bounded above by $\bar{U}$.Since we stipulate that agents always bid to get non-negative payoff, their utility will always be greater than or equal to that in last period. Thus the sum of all agents' utility, U_t, is nondecreasing. That is, ${{\partial U_t} \over {\partial t}} \geq 0$. Therefore the bounded increasing sequence $\{ U_t \}$ converges to a point U^*, such that $U^*=\lim_{t\rightarrow \infty}U_t$, and $U^*\leq \bar{U}$. $\Box$