Price Discrimination Based on Buyers’ Purchase History

We consider a repeated sales situation in which a seller sells one unit of a good each period to the same buyer. Having incomplete information about the buyer’s valuation, the seller responds to the buyer’s behavior, meaning, whether she bought previous units and at what price. We characterize the equilibrium of the no-commitment game for a uniform distribution and show that the seller discriminates between buyers based on their purchase history. Moreover, we show that the buyer’s ex-ante expected payoff is higher in the no-commitment game than in a game in which the seller can commit in advance to the prices he will charge (the commitment game). We also demonstrate that the buyer’s ex-post payoff is higher for each type of buyer in the no-commitment game. Social welfare is also higher in the no-commitment game. As the number of periods (units) increases, the opening prices decline, but fewer types of buyers accept the opening price.

1. Examples of other environments where units are sold one by one and if the buyer rejects the offer, the unit expires are rental contracts, markets for services, and markets for objects with an expiration date such as newspapers and tourism services.

2. This condition is exactly the regularity assumption in Myerson’s optimal auctions [25].

3. The proof of this result is standard; see, e.g., Fudenberg et al. ([14], Lemma 1), and Ausubel and Deneckere ([3], Lemma 2.1)

4. When the seller’s and buyer’s discount factors are the same, we can define the mechanism using only one price function \(P\left( v\right) \). However, when they are different, the mechanism will be given by \(\left\langle \left( p_{i}\left( v\right) ,q_{i}\left( v\right) \right) _{i=1}^{K}\right\rangle \) where \(p_{i}\left( v\right) \) is the price the buyer pays for unit i and \(q_{i}\left( v\right) \) is the probability that she will get unit i. Then the seller wants to maximize

   $$\begin{aligned} \int _{0}^{1}\left( \sum _{i=1}^{K}\delta _{S}^{i-1}p_{i}\left( v\right) \right) f\left( v\right) \mathrm{d}v \end{aligned}$$

   while the IC condition implies that

   $$\begin{aligned} \sum _{i=1}^{K}\delta _{B}^{i-1}\left( q_{i}\left( v\right) v-p_{i}\left( v\right) \right) \ge \sum _{i=1}^{K}\delta _{B}^{i-1}\left( q_{i}\left( \hat{v}\right) v-p_{i}\left( \hat{v}\right) \right) \end{aligned}$$

   In this case it is no longer true that the seller prefers to commit to non-contingent prices (we have constructed a counter example).

5. Note that \(Q\left( v\right) \) is non-differentiable at \(v=p^{*}\). However, as is common in the mechanism design literature, it is enough to assume that it is almost everywhere differentiable, which indeed holds. See, for example, the discussion on page 64 in [23].

6. By rearranging (17) we get

   $$\begin{aligned} v\left( 1-\sum _{i=1}^{K-j}\delta ^{i}\right) \le p_{j-1,K}-\delta p_{j,K}-c_{j,K}\sum _{i=2}^{K-j+1}\delta ^{i} \end{aligned}$$

   Recall that we assume that \(\delta \le \tilde{\delta }_{K}\), so the l.h.s is increasing in v. Moreover, (17) holds with an equality for \(v=c_{j-1,K}\).

Authors and Affiliations

1. Faculty of Business Administration, Ono Academic College, Kiryat Ono, Israel

   Arieh Gavious

2. Department of Industrial Engineering and Management, Ben-Gurion University, Beer-Sheva, Israel

   Arieh Gavious & Ella Segev

Corresponding author

Correspondence to Ella Segev.

Appendix: Proof of Proposition 1

We use the revelation principle and prove that the direct incentive compatible mechanism that maximizes the seller’s expected payoff is a mechanism in which the price is equal to the static monopoly price \(p^{*} \) in every period. This price is defined by the equation

Our assumption that \(p-\frac{1-F(p)}{f(p)}\) is strictly increasing ensures the existence and uniqueness of the solution of (3). Note that for \(p=1\), \(p-\frac{1-F(p)}{f(p)}>0\), while for \(p=0\), \(p-\frac{1-F(p)}{f(p)}<0\).

The direct bargaining mechanism here is a mechanism in which the buyer reports her valuation and then the mechanism determines what objects she will buy and how much she will pay. The price depends only on her type and not on how many objects she gets, because the price and allocation decisions are separate here. In other words, the direct mechanism is given by the price function \(P\left( v\right) \) and probability functions \(0\le q_{i}\left( v\right) \le 1\) that determine the probability that the buyer will get unit i when she reports v.^{Footnote 4} Therefore, for a direct mechanism \(\left\langle P\left( v\right) ,\left( q_{i}\left( v\right) \right) _{i=1}^{K}\right\rangle \) to be incentive compatible (IC) the following condition must hold

for any v and \(\hat{v}\ne v\). For such a mechanism to be individually rational (IR), the following condition must hold

for all \(v\in \left[ 0,1\right] \). The seller’s expected payoff in the mechanism \(\left\langle P\left( v\right) ,\left( q_{i}\left( v\right) \right) _{i=1}^{K}\right\rangle \) is given by

We show that the (IC) and (IR) direct mechanism that maximizes the seller’s expected payoff is given by

and

First note that this mechanism is indeed incentive compatible. If \(v<p^{*} \), the buyer’s expected payoff if the seller reports truthfully is zero. If she deviates and reports some other \(\hat{v}<p^{*}\), her expected payoff will remain the same. If she deviates and reports some \(\hat{v}\ge p^{*} \), her expected payoff is \(\sum _{i=1}^{K}\delta ^{i-1}v-p^{*}\sum _{i=1} ^{K}\delta ^{i-1}=\sum _{i=1}^{K}\delta ^{i-1}\left( v-p^{*}\right) <0\). If \(p^{*}\le v\le 1\), by deviating to some other \(p^{*}\le \hat{v}\le 1\) her expected payoff stays the same. If she deviates to some \(\hat{v}<p^{*} \), her expected payoff is zero and is (weakly) smaller than \(\sum _{i=1} ^{K}\delta ^{i-1}\left( v-p^{*}\right) \) for any \(p^{*}\le v\le 1\). Moreover, this mechanism is also individually rational. The seller’s expected payoff in this mechanism is given by

We show next that this is the (IC) and (IR) direct mechanism that maximizes the seller’s expected payoff. Assume that \(q_{i}\left( v\right) \) are differentiable and define

where \(0\le Q\left( v\right) \le \sum _{i=1}^{K}\delta ^{i-1}\). Define the buyer’s payoff given his type is v and he announces \(\hat{v}\) to be \(W(\hat{v}|v)=Q\left( \hat{v}\right) v-P\left( \hat{v}\right) \) and denote by \(W(v)=W(v|v)\) when the buyer reports her type truthfully. Then, the incentive compatibility condition implies that for every \(v\ge \hat{v}\)

By a similar argument \(W(v|\hat{v})\le W(\hat{v})\) and thus, we get \(\frac{W(v)-W(\hat{v})}{v-\hat{v}}\le Q(v)\). Thus, \(Q\left( \hat{v}\right) \le \frac{W(v)-W(\hat{v})}{v-\hat{v}}\le Q(v)\) and by taking the limit \(v\rightarrow \hat{v}\) we find that (we replace \(\hat{v}\) with v) \(\frac{d}{\mathrm{d}v}W(v)=Q(v)\) and by integration, \(W(v)=\int _{0}^{v}Q\left( s\right) \mathrm{d}s\). Since \(W(v)=Q\left( v\right) v-P\left( v\right) \), we find that

Note that individual rationality implies that when \(Q\left( v\right) =0\), \(P\left( v\right) =0\). We therefore we want to find the mechanism \(\left\langle P\left( v\right) ,Q\left( v\right) \right\rangle \) that maximizes

Integrating by parts we get

which is maximized when

and then, for \(p^{*}\le v\le 1\)

and for \(0\le v<p^{*},\, P\left( v\right) =0\).^{Footnote 5}

We know from the revelation principle that for any Bayesian Nash equilibrium of the commitment game there is an equivalent incentive compatible direct mechanism that always yields the same outcome (when the buyer reports truthfully). If the buyer’s strategy is to accept an offer of p each period iff \(v\ge p\), in the equivalent direct mechanism she will report her type truthfully. Thus, the best response for the seller is to determine all prices to be equal to the price \(p^{*}\). Moreover, the best response for the buyer is to buy whenever the price is below her valuation if all prices are equal to the price \(p^{*}\). Therefore, we conclude that in the Bayesian Nash equilibrium of the commitment game the seller’s strategy is to determine the price \(p^{*}\) for every period and after every history (and then \(P\left( v\right) \) is as defined above), while the buyer should accept an offer of p each period iff \(v\ge p\).

Proof of Theorem 1

The proof is by induction on the number of periods left, given K. The uniqueness is derived from the fact that at each step the solution to our maximization problem is unique. Based on [2] and by Fudenberg and Villas-Boas [16] we know that for \(K=2\) the seller’s strategy in the unique perfect Bayesian equilibrium is to offer

in the first period. All buyers such that \(v\in \left[ 0,c(p_{1})\right] \) where \(c(p_{1})=\frac{2}{2-\delta }p_{1}\) reject this offer, whereas all buyers such that \(v\in \left[ c(p_{1}),1\right] \) accept the offer. On the equilibrium path, we have \(c(p_{1}^{*})=\frac{2}{2-\delta }p_{1}^{*} =\frac{2+\delta }{4+\delta }>\frac{1}{2}\). In the second period, the seller’s belief is updated to \(\left[ c(p_{1}),1\right] \) if the buyer accepted the first offer and then he offers \(p_{2}=c\left( p_{1}\right) \) if \(c\left( p_{1}\right) \ge \frac{1}{2}\) and if \(c\left( p_{1}\right) <\frac{1}{2}\) he offers the price \(p_{2}=\frac{1}{2}\). If the buyer rejected the first period price, the seller updates his beliefs to \(\left[ 0,c\left( p_{1}\right) \right] \) and offers \(p_{2}=\frac{1}{2}c(p_{1})\) in the second period.

It is easy to verify that if \(\delta >0\), then \(U_{NC}^{B}\left( v,K=2\right) \ge U_{C}^{B}\left( v,K=2\right) \) for all \(v\in \left[ 0,1\right] \) and therefore also \(U_{NC}^{B}\left( K=2\right) \ge U_{C}^{B}\left( K=2\right) \), so the buyer prefers the no-commitment equilibrium ex-post as well as ex-ante. Finally, \(\mathrm{SW}_{NC}\left( K=2\right) \ge {rm SW}_{C}\left( K=2\right) \). Note that this equilibrium holds for every \(0<\delta \le 1\).

Preliminary Step

We first recursively define the prices and cutoffs.

We start with \(c_{K}^{*}=p_{K}^{*}\) and \(p_{K}^{*}=\frac{1}{2} c_{K-1}^{*}.\) Thus \(c_{K}^{*}=\frac{1}{2}c_{K-1}^{*}\). Assume that we already have expressions for \(c_{K}^{*},p_{K}^{*}, \ldots ,c_{j+1}^{*},p_{j+1}^{*}\) as a function of \(c_{j}^{*}\). The threshold \(c_{j}^{*}\) as a function of \(p_{j}^{*}\) is found by solving the following equation for c:

where it is understood that if we have a summation over indexes ranging from m to n and \(m>n\), the result of this summation is zero. Note that we assume here that if the seller observed rejections up to period j and then acceptance in period \(j+1\), the price in period \(j+2\) and onward would be equal to the cutoff \(c_{j+1}\). We prove below in Lemma (3) that this indeed holds in equilibrium.

Next, \(p_{j}^{*}\) as a function of \(c_{j-1}^{*}\) is found by plugging the expressions for \(c_{j}^{*}\) as a function of \(p_{j}^{*}\) into \(c_{K}^{*},p_{K}^{*}, \ldots ,c_{j+1}^{*},p_{j+1}^{*},c_{j}^{*}\) and solving

Finally, we again plug the expression \(p_{j}^{*}\left( c_{j-1}^{*}\right) \) into all previous expressions and get \(c_{K}^{*},p_{K}^{*}, \ldots ,c_{j}^{*},p_{j}^{*}\) as functions of \(c_{j-1}^{*}\). We repeat this step K times to find the values of \(c_{K}^{*},p_{K}^{*} , \ldots ,c_{1}^{*},p_{1}^{*}\) as functions of \(\delta \).

Our off the equilibrium path beliefs updating condition (condition 1) allows us to conclude that at any node in the game tree the seller’s beliefs can only be either of the form \(\left[ 0,d\right] \) for some \(d\ge 0\) or \(\left[ c,d\right] \) for \(c\ge \frac{1}{2}d\). Thus, we can describe the strategies as functions of these two possible kinds of beliefs. For this to be true, we only need to show that on the equilibrium path if the seller’s beliefs are of the form \(\left[ 0,d\right] \), they are updated to either \(\left[ 0,c\right] \) for some \(c\le d\), or to \(\left[ c,d\right] \) where \(c\ge \frac{1}{2}d\). We show this below where we prove that \(c_{i}^{*}\ge \frac{1}{2}c_{i-1}^{*}\) for all i. Once the seller’s beliefs are that the buyer’s valuation is distributed uniformly on some interval \(\left[ c,d\right] \) such that \(c\ge \frac{1}{2}d\), by our condition, after updating (either on or off the equilibrium path), the beliefs will still have this property. Finally, if the seller’s beliefs are of the form \(\left[ 0,d\right] \), both acceptance and rejection of the subsequent offer occur with positive probability. Therefore, there is never any updating of off the equilibrium path beliefs in this case and we can never end up with beliefs of the form \(\left[ c,d\right] \) such that \(c<\frac{1}{2}d\).

The Kth period

For simplicity of notation, hereafter we omit the superscript “\(*\)” from the equilibrium notations. We now explain the derivation of the cutoffs and prices for K periods while describing the players’ strategies in equilibrium. These strategies are a function of the history or equivalently a function of the seller’s beliefs. In the last period, the buyer’s best response is to accept an offer iff it is below her valuation, because this is a take-it-or-leave-it offer. Thus, for this period, the buyer’s strategy is given by

In this period the seller is making a take-it-or-leave-it offer given his beliefs. Thus

The \(K-1\)th period

If the seller’s beliefs are that the buyer’s valuation is distributed uniformly on an interval \([0,d]\), then the buyer’s strategy in the \(K-1\) period is

and the seller’s strategy is

If the seller’s beliefs are that the buyer’s valuation is distributed uniformly on an interval \([d_{0},d]\) such that \(d_{0} \ge \frac{1}{2}d\), then the buyer’s strategy is

and the seller’s strategy is

We explain how we derived these strategies for the \(K-1\) period.

Taking the strategy of the seller for the last period as given, if the seller’s beliefs are of the form \([0,d]\), then a buyer of type v accepts an offer p in period \(K-1\)th iff \(v-p+\delta \left( v-c\left( p\right) \right) \ge \delta \left( v-\frac{1}{2}c\left( p\right) \right) \) where \(c\left( p\right) \) is the type for which this inequality holds with an equality, i.e., \(c\left( p\right) =\frac{2}{2-\delta }p\). The l.h.s is her payoff if she accepts this offer, while the r.h.s is her payoff if she rejects it and waits for the next offer. For this equation to hold, we would need to show that \(c\left( p\right) \ge \frac{1}{2}d\) and then indeed the seller offers \(c\left( p\right) \) after an acceptance (when his beliefs are given by \(\left[ c\left( p\right) ,d\right] \)). Note that on the equilibrium path the seller updates his beliefs according to Bayes’ rule and the equilibrium strategy of the buyer. Thus, if the buyer rejects the offer, he updates his beliefs to \(\left[ 0,c\left( p\right) \right] \), and if she accepts the offer, he updates his beliefs to \(\left[ c\left( p\right) ,d\right] \).

Therefore, the seller responds best to the buyer’s behavior and maximizes his expected payoff for the rest of the game, i.e., for periods \(K-1\) and K, given his beliefs. The seller chooses p that maximizes

and thus

Note that indeed \(c\left( p\right) =\frac{2}{2-\delta }p=\frac{\left( \delta +2\right) }{\left( \delta +4\right) }d\ge \frac{1}{2}d\).

If the seller’s beliefs are of the form \([d_{0},d]\) such that \(d_{0}\ge \frac{1}{2}d\), then a buyer of type v accepts an offer p in period \(K-1\) iff \(v-p+\delta \left( v-c\left( p\right) \right) \ge \delta \left( v-d_{0}\right) \) where \(c\left( p\right) \) is the type for which this inequality holds with an equality, i.e., \(c\left( p\right) =\frac{p-\delta d_{0}}{1-\delta }\). If the buyer rejects the offer, then the seller updates his beliefs to \(\left[ d_{0},c\left( p\right) \right] \) and since \(d_{0}\ge \frac{1}{2}d\ge \frac{1}{2}c\left( p\right) \) he offers \(d_{0}\). Note that \(c\left( p\right) \ge d_{0}\Leftrightarrow p\ge d_{0}\). Otherwise, if the seller offers a price \(p<d_{0}\), then all buyers in the interval \([d_{0},d]\) will accept it and \(c\left( p\right) =d_{0}\). In this case, it is better for the seller to offer \(p=d_{0}\) and again all types in the interval \([d_{0},d]\) will accept it.

The seller therefore best responds to the buyer’s behavior and maximizes his expected payoff for the rest of the game given his beliefs. The seller chooses p that maximizes

and then

Thus, we have established the fact that if the seller’s beliefs are given by [c, d] such that \(c\ge \frac{1}{2}d\) and there are only two periods left the seller does not want to further screen buyers into high valuation and low valuation buyers and chooses the highest possible price offer that they all will accept.

Period \(j<K-1\)

We prove the theorem by induction on the number of periods left. Our induction hypothesis is as follows: 1. The buyer’s and seller’s strategies when fewer than \(K-j\) periods are left are the best responses to one another given the seller’s beliefs. 2. \(c_{j-1}\le c_{j-2}\le \cdots \le c_{0}=1\), and 3. If there are fewer than \(K-j\) periods left and the seller’s beliefs are given by \(\left[ c,d\right] \) such that \(c\ge \frac{1}{2}d\), in the subgame that follows the seller will offer c in each period and the buyer will accept. We now describe the \(\acute{j}\)’th period strategies as a function of the seller’s beliefs.

If the seller’s beliefs are given by \(\left[ 0,d\right] \), the buyer will accept the offer p iff it gives her a higher expected payoff in the subsequent subgame (given the strategies for periods \(j+1,j+2, \ldots ,K\)) than rejecting the offer, i.e., iff

where \(c\left( p\right) \) is the type for which this inequality holds with an equality.

The l.h.s is the buyer’s payoff if she accepts the current offer and then all subsequent offers (which will be equal to \(c\left( p\right) \)), and the r.h.s is her payoff if she rejects the offer and accepts the next offer and then all subsequent offers. If she accepts the offer, the seller updates his beliefs to \(\left[ c\left( p\right) ,d\right] \) and if \(c\left( p\right) \ge \frac{1}{2}d\) offers the price \(c\left( p\right) \) in the following period. In the subgame that follows, the buyer will accept all offers. Thus, we now have an expression for \(c_{j}(p)\) where \(c_{j}(p)<d=c_{j-1}(p)\). Note that if she decides to reject the offer and reject the next offer as well, her expected payoff is given by

and thus if \(v\ge c\left( p\right) \ge c_{j+1}\left( p\right) \). It follows by the induction that

because otherwise she would prefer to reject the offer \(p_{j+1}\), which contradicts the induction hypothesis that gave rise to the definition of \(c_{j+1}\left( p\right) \). This reasoning is true for any number of offers greater than 1. If she considers rejecting the offer, it is always best to accept the following offer, if \(v>c_{j+1}\left( p\right) \).

Thus, \(c_{j}\) as a function of p is given implicitly by:

and this condition also gives the complete definition of the buyer’s strategy (a buyer who rejected all offers up to \(j-1\)) for period j, given the seller’s beliefs. If the seller’s beliefs are given by \(\left[ 0,c_{j-1} \right] \), she accepts the offer p iff \(v\ge c_{j}\left( p\right) .\) Since we now have an expression for \(c_{j}\) as a function of p, we plug it into \(c_{j+1},\ldots , c_{K}\) and into \(p_{j+1}, \ldots ,p_{K}\) and express them all as functions of \(c_{j}\). Then we get that, in period j,  if the seller’s beliefs are given by \(\left[ 0,c_{j-1}\right] \), he chooses \(p\,\) that maximizes

Equations (4) and (5) therefore determine the buyer’s and seller’s strategies for period j given the seller’s beliefs, meaning, given \(c_{j-1}\), and their strategies in all following periods.

No further screening by the seller in period \(j<K-1\)

Lemma 3

If the seller’s beliefs in period j are given by \(\left[ d_{0},d\right] \) such that \(d_{0}\ge \frac{1}{2}d\), the seller sets \(p=d_{0}\)

Proof

If the seller’s beliefs in period j are given by \(\left[ d_{0},d\right] \) such that \(d_{0}\ge \frac{1}{2}d\), the buyer accepts the offer p iff

and \(v=c(p)\) is the type for which the last inequality holds with equality. Therefore,

The seller then chooses p that maximizes

Differentiating with respect to p we get

Note that to derive this result we use the fact that \(\sum _{i=1}^{K-j} \delta ^{i}\le 1\). We thus establish that if the seller’s beliefs are given by \([c_{i},c_{i-1}]\) such that \(c_{i}\ge \frac{1}{2}c_{i-1}\), the seller does not want to further screen buyers into high valuation and low valuation buyers and chooses the highest possible price offer that they will all accept \(p=c_{i}\). \(\square \)

A remark about the updating of beliefs is needed here. On the equilibrium path, updating is done using Bayes’ rule and is consistent with the buyer’s equilibrium strategy. Moreover, only the seller updates his beliefs (the buyer’s strategy is a function of the history and the current price). When the seller’s beliefs are given by \(\left[ 0,d\right] \), both accepting and rejecting the offer are possible equilibrium responses, and therefore the updating is again Bayesian. We thus conclude that the only possible updating off the equilibrium path is done when the seller’s beliefs are given by \(\left[ c,d\right] \) such that \(c\ge \frac{1}{2}d\) and he offers \(p=c\) and the buyer rejects the offer. Our Condition 1 then implies that the beliefs must be updated to an interval \(\left[ \tilde{c},\tilde{d}\right] \) in such a way that the property \(\tilde{c}\ge \frac{1}{2}\tilde{d}\) still holds, and the seller and buyer continue to behave optimally given these new beliefs. In the subgame that follows, the seller will offer \(\tilde{c}\) and the buyer will accept, where \(\tilde{c}\ge c\). Thus, if the seller’s beliefs are given by \(\left[ c,d\right] \) such that \(c\ge \frac{1}{2}d\) and he offers the price c, the buyer does not want to deviate and reject the offer. In this case, the seller does not want to deviate either, because \(p=c\) maximizes his expected payoff.

In order to complete the proof of the construction of the equilibrium strategies, all we must show is that \(c_{i}\ge \frac{1}{2}c_{i-1}\) for all i . To do so, we need the following results.

First, note that by the construction of the equilibrium we already know that \(c_{K}\le c_{K-1}\le \cdots \le c_{1}\le c_{0}=1\) and for all l we have \(p_{l}\le c_{l}\). We now prove that prices fall in equilibrium when the buyer turns down the offers, i.e., \(p_{K}=c_{K}\le p_{K-1}\le \cdots \le p_{2}\le p_{1}\)

Lemma 4

The equilibrium prices \(p_{j}\) decrease with j.

Proof

We want to prove that \(p_{j+1}\le p_{j}\). On the equilibrium path, after offering the price \(p_{j}\) the seller believes that the buyer’s valuation is distributed uniformly on the interval \(\left[ 0,c_{j}\right] \) if he received a rejection of his proposed price. We show that the seller’s next offer must be less than or equal to \(p_{j}\). Assume by negation that the seller offers \(p_{j}<p_{j+1}\le c_{j}\). We show that there exists a high valuation buyer (\(c_{j+1}\le v<c_{j}\)) in the interval \(\left[ 0,c_{j}\right] \) who would have been better off by accepting \(p_{j}\) - a contradiction. We therefore examine the expected payoff of such a buyer in both cases. If she rejects the offer \(p_{j}\), she will accept the offer \(p_{j+1}\) and her expected payoff is given by \(\delta \left( v-p_{j+1}\right) +\left( v-c_{j+1}\right) \sum _{i=2}^{K-j}\delta ^{i}\). If she accepts the offer \(p_{j}\), she will reject all subsequent offers equal to \(c_{j}\), because her valuation is below that price. Her expected payoff will therefore be \(v-p_{j}\). Now, since for \(\delta \le \tilde{\delta }_{k}\) we have

it follows that \(v-p_{j}>\delta \left( v-p_{j+1}\right) +\left( v-c_{j+1}\right) \sum _{i=2}^{K-j}\delta ^{i}\). Then, there exists v who accepts \(p_{j}\) - a contradiction. \(\square \)

We introduce the notations \(c_{j,K}\) and \(p_{j,K}\), which represent the prices and cutoffs for a given K. We then have specific relations between cutoffs and prices for a given K and prices and cutoffs for \(K-1\). The following Lemma is a constructive one. Starting from \(c_{1,1}=p_{1,1}=\frac{1}{2},\) it allows us to express \(p_{1,2},c_{2,2},p_{2,2}\) in terms of \(c_{1,2} \) and then, \(p_{1,3},c_{2,3},p_{2,3},c_{3,3},p_{3,3}\) all in terms of \(c_{1,2}\) and \(c_{1,3}\) and so on. Note that the range of \(\delta \) for which the equilibrium holds decreases with K. However, the prices and cutoffs for any number of periods are functions of \(\delta \) and can be written for any \(0<\delta \le 1\) and then the relations below hold. The fact that the range of \(\delta \) decreases means only that there might be a \(\delta \) for which \(c_{j-1,K-1}\) and \(p_{j-1,K-1}\) for \(j=2, \ldots ,K-1\) define an equilibrium for the \(K-1\) period game but the corresponding (according to the relations below) \(c_{j,K} \) and \(p_{j,K}\) for \(j=2, \ldots ,K\) do not.

Lemma 5

For all \(K\ge 2\), the following relations hold

where \(c_{1,1}=p_{1,1}=\frac{1}{2}.\)

Proof

Our equilibrium concept is a refinement of a perfect Bayesian equilibrium. Therefore, given a rejection in the first period, the cutoffs and prices of the remaining \(K-1\) periods must constitute a perfect Bayesian equilibrium of the subgame in which the seller’s initial belief is that the buyer’s valuation is distributed uniformly on the interval \(\left[ 0,c_{1,K}\right] \). This subgame is equivalent to the original game in \(K-1\) periods up to the multiplication by the constant \(c_{1,K}\). This is because beliefs are always updated to some truncated interval. This fact remains true when the initial belief is on the interval \(\left[ 0,c_{1,K}\right] \). Moreover, the equations that define the values of the cutoffs and prices (4) and (5) continue to hold if all of them are multiplied by \(c_{1,K}\). Therefore, the prices and cutoffs of the original game with \(K-1\) periods, multiplied by \(c_{1,K}\) constitute a perfect Bayesian equilibrium of the subgame that follows a rejection in the first period.

Thus, we have

Moreover \(p_{1,K}=\left( 1-\delta \left( 1-p_{1,K-1}\right) -\left( 1-c_{1,K-1}\right) \sum _{i=2}^{K-1}\delta ^{i}\right) c_{1,K}\). This follows directly from (4). \(\square \)

We will also use the following lemma:

Lemma 6

For \(K\ge 2\)

and

where it is understood that \(\sum _{i=m}^{n}b_{i}=0\) if \(m>n\) and \(c_{0,K}=1\) for all K and

Proof

For \(K\ge 2\), we know from (4) and (6) that (see the proof of Lemma 5)

and thus,

Then, by (5), \(p_{1,K}\) maximizes (recall that \(c_{0,K}=1\) for all K)

We note that by (6) we have the following relation

and therefore \(p_{1,K}\) maximizes

We now substitute \(c_{1,K}=\frac{1}{\left( 1-\sum _{i=1}^{K-1}\delta ^{i}+\delta p_{1,K-1}+c_{1,K-1}\sum _{i=2}^{K-1}\delta ^{i}\right) }p_{1,K}\). Then \(p_{1,K}\) maximizes the expression \(\left( 1-\frac{1}{\left( 1-\sum _{i=1}^{K-1}\delta ^{i}+\delta p_{1,K-1}+c_{1,K-1}\sum _{i=2}^{K-1} \delta ^{i}\right) }p_{1,K}\right) \) \(\left( 1+\frac{\sum _{i=1}^{K-1}\delta ^{i}}{\left( 1-\sum _{i=1}^{K-1}\delta ^{i}+\delta p_{1,K-1}+c_{1,K-1} \sum _{i=2}^{K-1}\delta ^{i}\right) }\right) p_{1,K} +\delta U_{NC}^{S}\left( K-1\right) \) \( \frac{1}{\left( 1-\sum _{i=1}^{K-1}\delta ^{i}+\delta p_{1,K-1}+c_{1,K-1}\sum _{i=2}^{K-1}\delta ^{i}\right) ^{2}}p_{1,K}^{2} \)

We derive it w.r.t. \(p_{1,K}\) and get the f.o.c.

Then

and

\(\square \)

We can now conclude that

Corollary 1

For all K and all \(j=2, \ldots ,K\) the following holds

Proof

We prove by induction on K. For \(K=2\) we know from (6) that \(c_{2,2}=c_{1,1}c_{1,2}\) and thus \(c_{2,2}=\frac{1}{2}c_{1,2}\). Assume we have established that for every n, \(n<K\), the inequalities \(c_{j,n}\ge \frac{1}{2}c_{j-1,n}\) for \(j=2, \ldots ,n\) hold. We want to prove for K. From (6) \(c_{j,K}\ge \frac{1}{2}c_{j-1,K}\Leftrightarrow c_{j-1,K-1}\ge \frac{1}{2}c_{j-2,K-1}\) which is true by induction for \(j=3, \ldots ,K-1\). Thus, we must prove that \(c_{2,K}\ge \frac{1}{2}c_{1,K}\) or equivalently that \(c_{1,K-1} \ge \frac{1}{2}\). This follows directly from Lemma 6. \(\square \)

We now prove one more lemma that we will use below.

Lemma 7

The first cutoff \(c_{1,K}\) is increasing with K, while the first price \(p_{1,K}\) is decreasing with K

Proof

We introduce another notation

Then, using this notation, by Lemma 6

Using these relations, we have the following recursive connection:

By (8),

Then we get

Therefore, \(c_{1,K}\) increases with K iff \(a_{K}\) increases with K. We first prove by induction that for all \(K\ge 1\)

We have \(a_{1}=\frac{1}{2}\delta \le \frac{1}{1-\delta }\left( \sqrt{1-\delta }-\left( 1-\delta \right) \right) \). Assume that \(a_{K-1}\le \frac{1}{1-\delta }\left( \sqrt{1-\delta }-\right. \left. \left( 1-\delta \right) \right) \). Then, since \(\frac{(1+x)^{2}}{2+x}\) is increasing with x, we have

Thus,

and the numerator is a polynomial of degree 2 in \(a_{K-1}\). It is easy to verify that all cases in which \(0\le a_{K-1}\le \frac{1}{1-\delta }\left( \sqrt{1-\delta }-\left( 1-\delta \right) \right) \) are nonnegative and therefore, \(a_{K}\) is increasing with K. Thus, by (14) \(c_{1,K}\) is also increasing with K.

All that remains is to show that \(p_{1,K}\) decreases with K. We first prove that for all \(K\ge 2\)

We prove this by induction on K. For \(K=2\) we have \(a_{2}=\frac{1}{2} \delta \frac{\left( \delta +2\right) ^{2}}{\delta +4}\) and \(a_{1}=\frac{1}{2}\delta \). Therefore, \(a_{2}-a_{1}= \frac{1}{2}\delta ^{2} \frac{\delta +3}{\delta +4}\), which is indeed strictly smaller than \(\delta ^{2}\frac{1+a_{1}}{2+a_{1}}=\delta ^{2}\frac{2+\delta }{4+\delta }\) for all \(0<\delta <1\). Assume by induction that \(a_{K-1}-a_{K-2}<\delta ^{K-1} \frac{1+a_{K-2}}{2+a_{K-2}}\) and observe that by the monotonicity of \(a_{K}\) we have \(\delta ^{K}\frac{1+a_{K-2}}{2+a_{K-2}}<\delta ^{K}\frac{1+a_{K-1} }{2+a_{K-1}}\). Then

Now, using (9) and (12) we have \(p_{1,K}=\frac{1}{\delta }\left( a_{K}-c_{1,K}\sum _{i=2}^{K}\delta ^{i}\right) \) and thus, by the monotonicity of \(c_{1,K}\), we have

\(\square \)

We use these lemmas to prove that no type of buyer wants to deviate from her equilibrium strategy. A buyer in the interval \(\left[ 0,c_{K,K}\right) \) who rejects all offers in equilibrium does not want to deviate. Since \(p_{j,K}\) decreases with j and \(p_{K,K}=c_{K,K}\), she will never accept any offer (her payoff from any such offer will be less than or equal to zero) and therefore she cannot achieve any positive payoff by deviating.

In equilibrium, a buyer \(v\in [c_{j,K},c_{j-1,K})\) for \(j=1, \ldots ,K\) rejects the first \(j-1\) offers and accepts the offer \(p_{j,K}\). Assume that \(j>1\). If a buyer \(v\in [c_{j,K},c_{j-1,K}]\) deviates and accepts an offer in some period \(l<j\). Her expected payoff from period l onward is given by \(v-p_{l,K}\) (she will never accept the subsequent offers of \(c_{l,K}\) since \(v<c_{j-1,K}\le c_{l,K}\)). We assume that the seller continues to follow his equilibrium strategy, because he cannot see that a deviation occurred. Therefore, he offers the price \(c_{l,K}\) in the subsequent period. We also assume that after deviating, the buyer follows her optimal strategy and thus refuses the offer \(c_{l,K}\). This leads to the off the equilibrium updating of beliefs by the seller. However, recall that no matter how he updates his beliefs, his subsequent offers will be greater than or equal to \(c_{l,K}\), so the buyer will reject them all.

The buyer’s expected payoff from period l onward if she acts according to the equilibrium strategy is, for \(2\le j\le K\)

We therefore have to prove that

For \(l=j-1\) this follows immediately from the definition of the indifferent type \(c_{j-1,K}\) in (4). For \(l=j-1\) and \(v=c_{j-1,K}\) (17) holds with equality and therefore is true for every \(v\le c_{j-1,K}\).^{Footnote 6}

We now show that she will consider such a deviation only in period \(l=\) \(j-2. \) In order to achieve this goal we prove that for \(K\ge 3\) and \(l=1, \ldots ,K-3\) and \(j=l+3, \ldots ,K\)

Therefore, a buyer \(v\in [c_{j,K},c_{j-1,K}]\) will never deviate and accept an offer in some period \(l\le j-3\), because the price is already higher than v in these earlier periods and her payoff is negative.

We prove the following:

Lemma 8

For \(K\ge 3,\, l=1, \ldots ,K-3\) and \(j=l+2, \ldots ,K-1\)

Proof

We prove by induction on K. First, note that for a given \(1\le l\le K-2\), it is sufficient to prove that \(p_{l,K}>c_{l+2,K}\), because \(c_{j,K}\) decreases with j. For \(K=3\) the only relation we have to prove is

By (2) we have:

Then

and this is easy to verify. We assume by induction that \(p_{l,K-1} >c_{l+2,K-1}\) for all \(l=1, \ldots ,K-4\) then, for K and \(l=2, \ldots ,K-3\) we have by (6):

and

Therefore, for \(l=2, \ldots ,K-3\) we use the induction assumption \(p_{l-1,K-1} >c_{l+1,K-1}\) and get the desired result. All that remains is to show that for all \(K\ge 3\)

Since \(c_{1,K}\) increases with K, we have \(c_{3,K}= c_{1,K-2} c_{1,K-1}c_{1,K}\le c_{1,K}^{3}\) and therefore it is sufficient to prove that

We know that \(p_{1,K}\) decreases with K and therefore \(p_{1,K}\ge \lim _{K\rightarrow \infty }p_{1,K}\) while \(c_{1,K}\) increases with K and therefore \(c_{1,K}^{3}\le \left( \lim _{K\rightarrow \infty }c_{1,K}\right) ^{3}\). In the limit, when \(K\rightarrow \infty \) we have, from Eq. (7), by abusing notations \(p_{1,\infty }=\lim _{K\rightarrow \infty }p_{1,K}\) and \(c_{1,\infty }=\lim _{K\rightarrow \infty }c_{1,K}\) that

Rearranging, we get

Now

and then, from Eq. (14)

and

Finally, for all \(\delta \le 0.523\,51\)

as desired. Note that \(0.523\,51>\tilde{\delta }_{5}=0.518\,79\). For \(K=2, 3, 4\), we can show directly that \(p_{1,K}>c_{1,K}^{3}\) \(\square \)

All that remains is to show that a buyer \(v\in [c_{j,K},c_{j-1,K}]\) will not deviate and accept an offer in period \(j-2\), or in other words, that (17) holds for \(l=j-2\). We do so in the following lemma:

Lemma 9

For any K and for any \(j=3, \ldots ,K\), a buyer \(v\in [c_{j,K},c_{j-1,K}] \) does not want to deviate and accept an offer in period \(j-2\). Formally,

Proof

We need to prove that for \(3\le j\le K\)

We already established that (17) holds for \(l=j-1\). By multiplying both sides by \(\delta \) we know that

Therefore, it is enough to prove that

To do so, it is sufficient to show that

Assume by contradiction that

Then

Moreover, since \(\delta \le \tilde{\delta }_{K}\) we have \(c_{j-2,K} -c_{j-1,K}\ge \left( \sum _{i=1}^{K-j+2}\delta ^{i}\right) \left( c_{j-2,K}-c_{j-1,K}\right) \) and

which contradicts (4) by which these values were defined. \(\square \)

We also need to consider another possible deviation in which a buyer \(v\in [c_{j.K},c_{j-1,K})\) accepts an offer in a period later than j, for \(j=1, \ldots ,K-1\). We again assume that the seller continues to act according to his equilibrium strategy. If this buyer then deviates and accepts an offer in some period \(l>j\), her expected payoff from period j onward is given by

because she will also accept all subsequent offers of \(c_{l,K}\).

We therefore prove the following lemma:

Lemma 10

For all K, all \(j=1, \ldots ,K-1\), all \(l=j+1, \ldots ,K\) and all \(v\in [c_{j,K},c_{j-1,K}]\) the following holds

Proof

Rearranging (19), we get

Therefore, it is enough to prove this inequality for \(v=c_{j,K}\). For \(l=j+1\) and \(v=c_{j,K}\) Eq. (20) holds with an equality by (4). Now, for \(j=1, \ldots ,K-2\) and \(l=j+2, \ldots ,K\) we need to show that

We prove this inequality by induction on K. For \(K=3\) we only need to prove that for \(j=1\) and \(l=3\)

which is easy to verify by (1, 2). Assume that it is true for \(K-1\), for all \(j=1, \ldots ,K-3\) and \(l=j+2, \ldots ,K-1.\)

Note that for \(j=2, \ldots ,K-2\) and \(l=j+2, \ldots ,K\) we can use (6) and then we need to prove

which follows immediately from the induction assumption. All that remains is to show that for \(j=1\) and \(l=3,\ldots ,K\)

From (4) we have

We use this equation to show that the r.h.s of (22) decreases with l (i.e., the most profitable forward deviation is the smallest \(l=3\)):

Therefore, we only need to show that for \(l=3,\)

or

We rearrange the l.h.s

and then use

and get that the l.h.s is equal to

while for the r.h.s

Using algebraic manipulations it is tedious but easy to verify that

\(\square \)

We argue that the fact that the seller does not want to deviate from his equilibrium strategy follows from the recursive construction of this strategy. In each period, given the continuation strategies and the updating of beliefs, the seller chooses the price that maximizes his payoff. This completes the proof of the theorem.

Proof of Theorem 2

We want to prove that

First, we observe that

or equivalently that

Now,

Therefore, we prove by induction on K that

For \(K=2\) we know that (see the beginning of Sect. 4)

Moreover, by Eq. (6)

By induction \(\sum _{i=1}^{K-1}\delta ^{i-1}c_{i,K-1}^{2}\le \frac{1}{4} \frac{1-\delta ^{K-1}}{1-\delta }\).

Then,

We then have to prove that

We already know that

where \(a_{j}=\delta p_{1,j}+c_{1,j}\sum _{i=2}^{j}\delta ^{i}\) for \(j=2,\ldots ,K\) and \(a_{1}=\frac{1}{2}\delta \). Since \(\frac{\left( 1+a_{K-1}\right) ^{2} }{\left( 2+a_{K-1}\right) ^{2}}\) increases with \(a_{K-1}\) and by the above, \(a_{K-1}\le \frac{1}{2}\delta \frac{1-\delta ^{K-1}}{1-\delta }\) then we have

Finally,

\(\square \)

Proof of Theorem 3

Since for a buyer of type \(v\le \frac{1}{2}\) the expected payoff in the commitment game is \(U_{C}^{B}\left( v,K\right) =0\), it follows immediately that \(U_{NC}^{B}\left( v,K\right) \ge U_{C}^{B}\left( v,K\right) \). We first show that for \(v\in \left[ c_{1,K},1\right] \) (recall that we already know that \(c_{1,K}\ge \frac{1}{2}\) from corollary 1) the result holds. For \(v\in \left[ c_{1,K},1\right] \) we need to show that

Rearranging gives

Recall that (see 11)

and thus, we need to prove that

However, we already know that

All that remains is to show that for \(v\in [1/2,c_{1,K}]\) the result holds. We first prove for \(K=2\). For \(K=2\) we have \(c_{2,2}=\frac{1}{2} c_{1,2}\) and \(c_{2,2}\le \frac{1}{2}\). Therefore, \(v\ge \frac{1}{2}\) iff \(v\ge c_{2,2}\). For \(v\in \left[ c_{1,2},1\right] \) we already know that the theorem holds. Therefore, all we must do is show that it holds for \(v\in \left[ \frac{1}{2},c_{1,2}\right] \subset \left[ c_{2,2},c_{1,2} \right] \). Then,

but

and since \(c_{1,2}=\frac{2+\delta }{4+\delta }\) this inequality is true.

We prove that \(c_{2,K}\le \frac{1}{2}\) for all \(K>2\). Then, for any such type \(v\in [1/2,c_{1,K}]\) it holds that \(v\in [c_{2,K},c_{1,K}] \). Note that by Lemma 5 \(c_{2,K}=c_{1,K-1}\cdot c_{1,K}\). In Lemma 7 we show that \(c_{1,K}\) is increasing in K and thus \(c_{1,K-1}\) is also increasing in K and \(c_{2,K} \) is increasing in K as well. Therefore, \(c_{2,K}\le c_{2,\infty } \). Note again that when K grows, the range of the parameter \(\delta \) for which the equilibrium holds decreases. However, for any K, we write the expression for \(c_{2,K}\) as a function of \(\delta \) for any \(0<\delta \le 1\). This is what we mean when we claim that \(c_{2,K}\) is increasing in K for a given \(\delta \). Moreover, when K goes to infinity, we have (by 18)

It follows that \(c_{2,\infty }\le 1/2\) for every \(\delta \le \) \(2\sqrt{2}-2= 0.828\,43\) that holds under our assumption for \(K>2\) (for \(K>2 \), we have \(\tilde{\delta }_{K}<2\sqrt{2}-2\). However, we are interested only in those expressions for \(c_{2,K}\) when \(\delta \le \) \(\tilde{\delta }_{K} \)) . All that remains is to show that all types who in equilibrium reject the first offer and accept the second one and whose valuation is greater than 1 / 2 will prefer the no-commitment game.

Thus, for \(v\in [1/2,c_{1,K}]\) we need to show that \(U_{NC}^{B}\left( v,K\right) \ge U_{C}^{B}\left( v,K\right) \)

Then, it is sufficient to prove that

Note that \(\frac{1}{2}\left( \sum _{i=0}^{K-1}\delta ^{i}\right) -\delta p_{2,K}-c_{2,K}\sum _{i=2}^{K-1}\delta ^{i}\!=\!\frac{1}{2}\left( \frac{1-\delta ^{K}}{1-\delta }\right) -\left( \delta p_{1,K-1}\!+\! c_{1,K-1}\sum _{i=2}^{K-1}\delta ^{i}\right) c_{1,K}=\frac{1}{2}\left( \frac{1-\delta ^{K} }{1-\delta }\right) -a_{K-1}c_{1,K}\).

Therefore, we need to prove that

By (14)

Thus, we need to show that

The l.h.s is increasing in \(a_{K-1}\) and we know from the proof of Theorem 2 that

Then,

Finally,

which is indeed true for all \(0<\delta \le 1\). \(\square \)

Power Distributions

We solve for the equilibrium behavior with a general power distribution \(F\left( v\right) =v^{\alpha }\) for any \(\alpha >0\). For \(K=3\), in the third period, given that the seller believes that v is distributed on \(\left[ 0,c_{2}\right] \) according to the truncated distribution \(F_{c_{2}}\left( v\right) =\left( \frac{v}{c_{2}}\right) ^{\alpha }\) , he offers the price p that maximizes \(p\left( 1-F_{c_{2}}\left( p\right) \right) \) and therefore \(p_{3}=c_{3}=\frac{1}{\left( \alpha +1\right) ^{\frac{1}{\alpha }} }c_{2}\). We omit the rest of the calculation of the cutoffs and prices and present the expression for the seller’s expected payoff next:

We can show numerically (by simulations on thousands of different values for \(\delta \) and \(\alpha \)) that it is indeed always smaller than

and moreover

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