Reverse Auction bidding strategy

A buyer wants to buy contiguous land plots from three landowners (sellers) who own fixed size plots. Sellers have their private valuation of their plots. The buyer will hold a closed bid procurement auction. The profit for the buyer is greater if he purchases all plots of land as compared to (any) 2 plots(without any specific order), which itself is greater than profit for a single plot. Does someone have an idea of how strategies will be developed for the sellers (in equilibrium)? I will later extend it to multiple rounds reverse auctions.

• Can you be a bit more specific on the setting of the sellers and the valuation by the buyer? E.g. does the buyer only value 2 plots more if they are adjacent or are plots always adjacent? How does size of the individual plots affect the value? Does every seller own a single plot or multiple ones? – Maarten Punt Nov 13 '17 at 15:33
• @MaartenPunt all plots are always adjacent and every seller owns only a single plot – sujeet14108 Nov 13 '17 at 15:59

Given the very general description of the problem, I can think of the following (also very general) way of formulating it mathematically.

Let $v_n$ be the buyer's value from owning $n$ plots of land, with $v_3>v_2>v_1$. Let $c_i$ be seller $i$'s private valuation of the plot she owns. Let $b_i$ be seller $i$'s bid, and we assume the buyer will purchase $n$ plots of land whenever $v_n$ is greater than or equal to the sum of the $n$ smallest $b_i$'s, and each seller gets the amount she asks for in the bid.

Seller 1's objective is to choose $b_1$ to maximize the following: \begin{align} &\Pr(b_1+b_2+b_3\le v_3)b_1 \\&\qquad+\Pr(b_1\le\max\{b_1,b_2,b_3\}\text{ and }b_1+\min\{b_2,b_3\}\le v_2\text{ and }b_1+b_2+b_3> v_3)b_1 \\&\qquad+\Pr(b_1=\min\{b_1,b_2,b_3\}\text{ and }b_1\le v_1\text{ and }b_1+\min\{b_2,b_3\}> v_2)b_1 \end{align} The problems for the other two sellers would be symmetric.

If you assume that $c_i$'s are i.i.d. draws from a distribution $F$, then it seems plausible to solve for a symmetric Bayesian Nash equilibrium, where every seller bids according to the same bidding function, $b_i=\beta(c_i)$ for each $i$.

Using $b_i=\beta(c_i)$, and perhaps with the assumption that it is strictly increasing, you should be able simplify the $\Pr(\cdot)$'s in the objective function. The rest is just to solve for the bidding function. I will leave you to finish the remaining steps.

• Can you please solve the model further. I am not able to model the probabilities in the form F. – Mukesh Gupta Nov 14 '17 at 11:40
• @MukeshGupta I don't want to do your work for you. But you can read up on finding symmetric equilibrium in first price auction, since the remaining steps will be similar. A good source will be Krishna's Auction Theory. – Herr K. Nov 14 '17 at 14:07
• In the book you recommended , the buyer with maximum bid wins. There is not constraints like valuation limits and dynamic selection of multiple bids. I am able to simply the max functions, but how should I go with P(b1 + b2 + b3 < v3) ? – Mukesh Gupta Nov 14 '17 at 16:54