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I've found an interesting question looking at perfect-bayesian-equilibrium. I haven't seen a question where beliefs are not discrete.

There is a single potential buyer of an object which has zero value to the seller. This buyer’s valuation v is uniformly distributed on [0, 1] and is private information. The seller names a price $p_1$ which the buyer accepts or rejects.

If he accepts, the object is traded at the agreed price and the buyer’s payoff is $v − p_1$ and the seller’s is $p_1$.

If he rejects then the seller makes another price offer, p2. If the buyer accepts this, his payoff is $\delta_(v − p_2)$ and the seller’s is $\delta p_2$, where $\delta = 0.5$.

If he rejects, both players get zero (there are no further offers).

Find a Perfect Bayesian Equilibrium.

My usual approach is to fix beliefs, but I don't quite know how to do this with continuous beliefs. Any advice?

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  • $\begingroup$ Sorry, I could not think of an easy way to give partial advice. This is a nice exercise. Would you (or the creator) mind if I used it in class? $\endgroup$ – Giskard May 2 '15 at 21:51
  • $\begingroup$ Of course, feel free! $\endgroup$ – Brian May 3 '15 at 10:52
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After posting a bad solution yesterday I believe I got a better one:

The strategy of the buyer consists of two functions, $(f_1(v,p_1),f_2(v,p_1,p_2))$ where both functions map to $\left\{A,R\right\}$ (where $A$ stands for Accept, $R$ for Reject). The strategy of the seller is $(p_1,p_2(f_1(v,p_1)))$. You get the solution via backward induction. In PBE $f_2(v,p_1,p_2)$ maps to $A$ if and only if $v \geq p_2$. (There is inconsequential leeway at equality.) In PBE the seller believes that there is a set $H$ of types for which the buyer refused her offer $p_1$. Then $$ p_2^* = \arg\max_{p_2} p_2 \cdot Prob(f_2(v,p_1,p_2) = A | f_1(v,p_1) = R). $$ The buyer will accept offer $p_1$ if and only if $$ v - p_1 \geq \delta \cdot (v - p_2). $$ From this you get $$ v \cdot (1 - \delta) \geq p_1 - \delta \cdot p_2. $$ The left hand side of this equation is increasing in $v$, so types with high valuation will Accept. This means that in PBE the set $H$ is such that $$ H = [0, \bar{v}). $$ From this we get the optimal $p_2$ given $\bar{v}$: $$ p_2^* = \arg\max_{p_2} p_2 \cdot Prob(v \geq p_2 | v \in [0, \bar{v})) = \frac{\bar{v}}{2}. $$ In PBE $\bar{v}$ is a function of $p_1$: $$ \bar{v} \cdot (1 - \delta) = p_1 - \delta \cdot \frac{\bar{v}}{2}, $$ so $$ \bar{v} = \frac{p_1}{1 - \frac{\delta}{2}}. $$ We have determined all the PBE strategies but $p_1$. The expected payoff of the seller is $$ p_1 \cdot \left( 1 - \frac{p_1 - \delta \cdot p_2(\bar{v}(p_1))}{1 - \delta} \right) + \frac{1}{2} \cdot p_2(\bar{v}(p_1)) \cdot \left( \frac{p_1 - \delta \cdot p_2(\bar{v}(p_1))}{1 - \delta} - p_2(\bar{v}(p_1)) \right), $$ where $$ p_2(\bar{v}(p_1)) = \frac{\bar{v}(p_1)}{2} = \frac{\frac{p_1}{1 - \frac{\delta}{2}}}{2} = \frac{p_1}{2 - \delta}. $$ Substituting this we get $$ p_1 \cdot \left( 1 - \frac{p_1 - \delta \cdot \frac{p_1}{2 - \delta}}{1 - \delta} \right) + \frac{1}{2} \cdot \frac{p_1}{2 - \delta} \cdot \left( \frac{p_1 - \delta \cdot \frac{p_1}{2 - \delta}}{1 - \delta} - \frac{p_1}{2 - \delta} \right), $$

You have to maximize this w.r.t. $p_1$. With $\delta = 0.5$ I got $$ p_1^* = \frac{9}{20}, \hskip 20pt \bar{v} = \frac{3}{5}, \hskip 20pt p_2^* = \frac{3}{10}. $$

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  • $\begingroup$ I feel like this question can also be interpreted as a firm trying to screening consumers of different valuations represented as the closed unit interval. The optimal pricing scheme is to set two prices so that customers of high valuations will pay at a higher price at first stage, and some of those of low valuations will pay at a lower price at second stage. $\endgroup$ – Metta World Peace May 3 '15 at 14:48
  • $\begingroup$ You have to explain why the utilities are different in round 2. For the seller it could be simple discounting, but for the buyer? If the good were durable then types that buy the good would receive some benefits in both rounds. $\endgroup$ – Giskard May 3 '15 at 15:07
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    $\begingroup$ I don't quite follow. Why the buyers can't discount the utility derived in second round? This can be interpreted as a two-period price skimming, right? $\endgroup$ – Metta World Peace May 3 '15 at 15:36
  • $\begingroup$ Embarrassing but I have never heard of this model until now. You are correct, this describes the above game nicely. $\endgroup$ – Giskard May 3 '15 at 17:30
  • $\begingroup$ You said that the buyer will accept $p_1$ if and only if $$v-p_1\ge \delta(v-p_2)$$ but won’t the buyer reject if both $p_1$ and $p_2$ are greater than $v$, regardless of whether the above inequality is satisfied? $\endgroup$ – Franklin Pezzuti Dyer Apr 14 at 14:41

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