<blockquote>
  <p>Consider a seller offering a single object for sale to two buyers with
  independent valuations – each bidder’s value is uniformly distributed
  on [0,1]. Assume that buyer 1 submits a bid that buyer 2 observes.
  Buyer 2 then has the ‘first refusal’ right to buy the object at that
  price. If she refuses, buyer 1 gets the object at the price he bid. What is the optimal strategy for buyer 1?</p>
</blockquote>

<p>I have attempted the following:</p>

<p>$$\pi_1(v_1,v_2)=p(v_2&lt;\beta(v_1))(v_1-\beta(v_1))$$
$$=(1-\beta(v_1))(v_1-\beta(v_1))$$</p>

<p>If this is an equilibrium, we must have that</p>

<p>$$=(1-\beta(v_1))(v_1-\beta(v_1)) \geq (1-\beta(w))(v_1-\beta(w))$$</p>

<p>for all $w\in[0,1]$. Hence,</p>

<p>$$\left.\frac{\partial \pi_1(v_1,v_2,w)}{\partial w}\right|_{w=v_1}=-v_1\beta'(v_1)-\beta'(v_1)+2\beta(v_1)\beta'(v_1)=0$$</p>

<p>$$\iff\beta(v_1)=\frac{1+v_1}{2}$$</p>

<p>But this strikes me as clearly wrong, if $v_1=0$ why should the player overbid? What am I missing?</p>

<game-theory><auctions>
Bayes-Nash equilibrium in Auction
user526463
# Bayes-Nash equilibrium in Auction

(Asked by: username_0 on 2018-04-04)

> Consider a seller offering a single object for sale to two buyers with
> independent valuations – each bidder’s value is uniformly distributed
> on [0,1]. Assume that buyer 1 submits a bid that buyer 2 observes.
> Buyer 2 then has the ‘first refusal’ right to buy the object at that
> price. If she refuses, buyer 1 gets the object at the price he bid. What is the optimal strategy for buyer 1?

I have attempted the following:

$$\pi\_1(v\_1,v\_2)=p(v\_2<\beta(v\_1))(v\_1-\beta(v\_1))$$
$$=(1-\beta(v\_1))(v\_1-\beta(v\_1))$$

If this is an equilibrium, we must have that

$$=(1-\beta(v\_1))(v\_1-\beta(v\_1)) \geq (1-\beta(w))(v\_1-\beta(w))$$

for all $w\in[0,1]$. Hence,

$$\left.\frac{\partial \pi\_1(v\_1,v\_2,w)}{\partial w}\right|\_{w=v\_1}=-v\_1\beta'(v\_1)-\beta'(v\_1)+2\beta(v\_1)\beta'(v\_1)=0$$

$$\iff\beta(v\_1)=\frac{1+v\_1}{2}$$

But this strikes me as clearly wrong, if $v\_1=0$ why should the player overbid? What am I missing?

**Tags:** game-theory, auctions

---

## 0 answers

