Best-response bidding

Let agent i be the learning agent, with reservation prices $\bar{P}_{b}^i$ and $\bar{P}_{s}^i$. Let $\{\hat{P}_{b}^{1},\ldots, \hat{P}_{b}^{n}\}$ and $\{\hat{P}_{s}^{1},\ldots, \hat{P}_{s}^{n}\}$ be agent i's projected buying and selling prices of other agents. Suppose agent i can be matched as a buyer according to its reservation price, and is matched to seller j. By the matching criterion, we must have $\bar{P}_{b}^i \geq \hat{P}_{s}^{j}$. From the discussion before, we know that agent i wants to bid a buy price lower than its own reservation buy price, i.e. $P_{b}^i \leq \bar{P}_{b}$, and indeed should bid as low as possible in order to maximize its payoff. However, if it bids lower than , it will lose the chance to trade with agent j.

  

Figure 3: Choose best-response bid

Another way agent i might lose the opportunity to trade is if its buy price is lower than the other potential buyers. Suppose that agent k has the next highest bid other than agent i. If $\hat{P}_{b}^{k} \geq P_{b}^i$, agent k will get the match with agent j.

Therefore, agent i maximizes its reward by choosing

$$P_{b}^i=\max \{\hat{P}_{s}^{j}, \hat{P}_{b}^{k}+\epsilon\},$$ (6)

where is a small positive constant representing the minimal bid increment. Note that the above equation automatically satisfies the condition $P_{b}^i \leq \bar{P}_{b}$ because $\hat{P}_{s}^{j} \leq \bar{P}_{b}^i$ and $\hat{P}_{b}^{k} \leq \bar{P}_{b}^i$.

By similar analysis, suppose agent i can be matched as a seller to buyer j, and the next lowest seller is agent k, agent i should choose its sell-price such that

$$P_s^i=\min \{ \hat{P}_b^{j}, \hat{P}_s^{k}-\epsilon\}.$$

potential sellers to be matched with agent i's P^s,r.

If agent i predicts that it cannot be matched either as a buyer or a seller according to its reservation prices, it simply bids its reservation prices.