Three types of agents

 We designed three types of agents. One is competitive agents who always bid their true reservation prices. The other two are strategic learning agents who choose their bidding prices based on their possible influence on the market. They include price modeling agents and agents who model the bidding of other agents.

Let $\bar{P}^{b}$ and $\bar{P}^{s}$ be an agent's reservation buy and sell prices. The best sell bid that an agent can submit is $\bar{P}^{s}$. This is because Mth-price auctions are incentive compatible for a seller, given that the seller considers only one period's payoff [Wurman, Walsh, & Wellman1998].

A price modeling agent looks at the history data of clearing prices, and predicts the next clearing price. It estimates a time series model,

$$P_t=\alpha P_{t-1}+\varepsilon.$$

After predicting the next clearing price P_t, which is the Mth price in the auction, the agent then choose its best response buy bid such that

$$P^{b}=\min \{\bar{P}^b, P_t-\delta\},$$ (2)

where $\delta$ is a predefined constant which reflects the greediness of the agent.

A bidder-modeling agent models the actions of other agents by looking at the history data of those actions, and uses time series techniques to predict the actions in the next time period. For any other agent k, the bidder-modeling agent predicts agent k's bid in the next period, P_t^k, by  

$$P_t^k=\beta P_{t-1}^k+\varepsilon$$ (3)

where $\beta$ is a parameter estimated from agent k's price history.

After forming predictions of other agents' bids, the strategic agent chooses its new bids as a best response to these estimates.

Let $\{\hat{P}_{b}^{1},\ldots, \hat{P}_{b}^{n}\}$ and $\{\hat{P}_{s}^{1},\ldots, \hat{P}_{s}^{n}\}$ be the strategic agent's projected buy and sell prices of other agents. Let P_M be the predicted Mth price, and P_M+1 be the predicted M+1st price. If $\bar{P}^b<P_M$, the agent cannot be matched as a buyer then it does not matter what bid it submits. Since the agent has uncertainty about the actual bids in the market, the best bid it can submit is its reservation price $\bar{P}^{b}$. If $\bar{P}^b\geq P_M$, the agent wants to reduce the Mth price so that it can make more profit. The way to do this is to submit a price P^b that is lower than P_M but higher than P_M+1 so that this price will become the Mth price.

  

Figure: Choose best-response bid

Therefore, the agent's best response buy bid is

$$P^{b}=\min \{\bar{P}^b, P_{M+1}+\epsilon\},$$ (4)

where is a small positive constant representing the minimal bid increment. Note that the above equation automatically satisfies the condition $P^{b} \leq \bar{P}_{b}$ which means that the agent's utility will never decrease.