Modeling the auction

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

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

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 our agent design, we let $\alpha_g =1$ for all g, and $\rho = {1 \over 2}$, and so the utility function becomes $U(x)=\biggl(\sum _g (x_g)^{1 \over 2}\biggr)^2$.

The reward for agent i at time t is given by

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 [Varian1992]. The reservation price is defined as the maximum (minimum) price an agent is willing to pay 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 stays constant when buying or selling one unit of a good.

For example, in the case of one good, g, and money, m, the reservation buy price, $\bar{P}_{b}$, is the price such that the current utility is equal to the utility with one additional good of type g (the one we would be buying) and reduction in money of $\bar{P}_{b}$(the price we would pay). In other words, since the utility remains constant, an agent would be indifferent to making such a transaction. At any lower price than the reservation buy price, the agent will increase its utility by transacting.

It is shown in [Hu & Wellman1998] that for quasi-concave (such as CES) utility, the agent's reservation buy price is always lower than its reservation sell price.

  

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