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

<p>Let $v_n$ be the buyer's value from owning $n$ plots of land, with $v_3&gt;v_2&gt;v_1$. 
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. </p>

<p>Seller 1's objective is to choose $b_1$ to maximize the following: 
\begin{align}
&amp;\Pr(b_1+b_2+b_3\le v_3)b_1
\\&amp;\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&gt; v_3)b_1
\\&amp;\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\}&gt; v_2)b_1
\end{align}
The problems for the other two sellers would be symmetric. </p>

<p>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 $i$. </p>

<p>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. </p>

CC BY-SA 3.0
2017-11-14T11:40:49.147Z
Can you please solve the model further. I am not able to model the probabilities in the form F.
Mukesh Gupta
CC BY-SA 3.0
2017-11-14T14:07:11.307Z
@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.
CC BY-SA 3.0
2017-11-14T16:54:41.110Z
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
CC BY-SA 3.0
2017-11-14T02:46:52.830Z
2017-11-14T02:46:52.830Z
Herr K.
<p>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. </p>

CC BY-SA 3.0
2017-11-13T15:33:29.937Z
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
CC BY-SA 3.0
2017-11-13T15:59:43.863Z
@MaartenPunt all plots are always adjacent and every seller owns only a single plot
sujeet14108
<game-theory><auctions>
Reverse Auction bidding strategy
sujeet14108
# Reverse Auction bidding strategy

(Asked by: username_0 on 2017-11-13)

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.

**Tags:** game-theory, auctions
> **Comments:**
> **username_1**: 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?  
> **username_0**: @MaartenPunt all plots are always adjacent and every seller owns only a single plot  

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## 1 answers

### Answer by username_2 on 2017-11-14. Score: 5

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.

> **Comments:**
> **username_3**: Can you please solve the model further. I am not able to model the probabilities in the form F.  
> **username_2**: @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*.  
> **username_3**: 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) ?  

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