# Dynamic Pricing Decisions and Seller-Buyer Interactions under Capacity Constraints

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. The Model

_{1}that is announced to all the consumers. Consumers then respond independently by choosing whether to attempt purchasing at that price. If the demand exceeds the inventory, then a proportional rationing scheme [38] is carried out so that each consumer who expresses her wish to purchase is assigned the good with equal probability; this scenario is effectively equivalent to the assumption that consumers arrive at the market randomly throughout the period and the first ones get to purchase the good on a first-come, first-served basis. A consumer with valuation v gains a net payoff of v-p

_{1}if she purchases the good in period 1 at the price p

_{1}, after which she leaves the market. If the entire inventory is sold in period 1, then the seller leaves the market. Otherwise, at the beginning of period 2 the seller announces a price p

_{2}, and the consumers who have not purchased the good in period 1 decide independently and simultaneously whether to attempt purchasing it at that price. Similar to period 1, a proportional rationing scheme is implemented if more consumers attempt to purchase the good than the remaining inventory.

_{1}, the seller determines in part the proportion of consumers who postpone their purchase to period 2; in doing so, she has to estimate the proportion of consumers who behave strategically in response to p

_{1}and in expectation of what the period 2 price would be.

_{2}. When $\delta $ approaches zero, every consumer becomes effectively a myopic buyer, who would purchase in period 1 if her valuation is higher than the current price; such a buyer is also non-strategic, as she effectively does not look forward to possible future changes in price when making purchase decisions. The seller also has time preference over profits (due to the cost of capital and other factors) with time discount factor ${\delta}_{F}\in (0,1)$. We assume that both time discount factors are common knowledge, and that ${\delta}_{F}$ ≥ $\delta $, i.e., that the seller’s time preference over profits is not stronger than the consumer’s time preference over payoff. This assumption nests two commonly modeled scenarios, namely, ${\delta}_{F}$ = 1 (controlling for the profit, the seller is indifferent between earning it in either period), and ${\delta}_{F}$ = $\delta $ (the seller has the same time discount factor as the consumers) at two ends of a continuum. Moreover, the seasonal nature of the good means that its value to the consumers effectively depreciates to zero at the end of the game, while profits earned by the seller are not expected to be discounted as strongly.

_{1}× (number of consumers who successfully purchase the good in period 1)

+ δ

_{F}p

_{2}× (number of consumers who successfully purchase the good in period 2).

#### Equilibrium

**Proposition**

**1.**

- (i)
- ${I}_{1}=\frac{1-{\delta}_{F}}{2-\delta -{\delta}_{F}}$;
- (ii)
- ${I}_{2}=$ the larger root of the equation $G(I;\delta ,{\delta}_{F})=0$, where :$$\begin{array}{ll}G(I;\delta ,{\delta}_{F})\equiv & -[4{\delta}_{F}-{({\delta}_{F}+\delta )}^{2}]{I}^{2}+2(3{\delta}_{F}-{\delta}_{F}^{2}-\delta {\delta}_{F}-\delta )I\\ \text{}& \text{\hspace{1em}}-\frac{3-\delta -{\delta}_{F}}{4-2\delta -{\delta}_{F}}\left[(3{\delta}_{F}-{\delta}_{F}^{2}-\delta {\delta}_{F}-\delta )-\delta (1-\delta )\right]\text{\hspace{0.17em}};\end{array}$$

- (1)
- When I ≤ ${I}_{1}$, the seller does not attempt price skimming in equilibrium, and the entire inventory is sold in period 1 at the price of 1-I (called “one-period equilibrium”);
- (2)
- When ${I}_{1}$ < I ≤ ${I}_{2}$, selling takes place over both periods in equilibrium, there is some form of price skimming, and the entire inventory is sold over both periods (called “Type I two-period equilibrium”);
- (3)
- When ${I}_{2}$< I, selling also takes place over both periods in equilibrium with price skimming, but some inventory is left unsold after period 2 (called “Type II two-period equilibrium”).

_{2}, when there are two different equilibria, one of Type I two-period and one of Type II two-period, that yield the same profit.

_{2}—this quantity only demarcates when, as inventory increases, a Type II two-period type of pricing strategy becomes an equilibrium strategy.

## 4. The Experiment

#### 4.1. Subjects

#### 4.2. Design

_{F}= δ = 0.5).

#### 4.3. Procedure

- (1)
- If fewer than I Buyers made a purchase decision, then the round would proceed to period 2;
- (2)
- If exactly I Buyers made a purchase decision, then the round would be over;
- (3)
- If more than I Buyers made a purchase decision, then exactly I Buyers would randomly be chosen to purchase the laboratory good and the round would be over.

#### 4.4. Equilibrium Predictions and Choice of Conditions

## 5. Results

#### 5.1. Preliminary Analysis: Aggregate Decisions Largely Approximate Equilibrium

#### 5.2. Analysis of Pricing Decisions: Boundedly Strategic Behavior among the Sellers

#### 5.3. Systematic Deviations in Demand

#### 5.4. The Relationship between Deviations in Demand and the Ex-Post Optimal Price Offer

_{1}, v being her valuation and v** being the cutoff valuation for best response purchases as defined for Table 2. That is, the buyer should not have attempted to purchase even though her valuation is higher than the period 1 price, since the equilibrium prescribed that the buyer should wait strategically as a best response; myopic buying occurred when this occasion arose and the buyer “attempted purchase myopically” vis a vis considerations of rational expectations.

#### 5.5. Learning Analysis

_{1}as discussed previously); we then carry out a logistic regression over these instances for each condition, with the likelihood of myopic buying as the dependent variable and the period 1 price, the round number, and the interaction term (period 1 price × round number) as the three independent variables. We use the generalized estimating equations (GEE; see [41]) approach in the regression to account for possible dependencies among observations from the same subject or the same session. As it turns out, in Condition I9 the estimated coefficient (standard error in parentheses) for round number is −0.40 (0.16), and that for the interaction term is 0.0022 (0.0009); both coefficients are significantly different from zero at p < 0.05. Note that, except in rounds where period 1 prices were higher than 180 (which constitute only 17 of 315 observations), the positive interaction effect could not overturn the negative round number effect; hence, in general, the buyers were less prone to myopic behavior as the session progressed, controlling for the period 1 price. The corresponding estimations for Condition I16 are not significantly different from zero (p > 0.8 for both coefficients).

## 6. Conclusions

- Period 1 Seller’s prices and Buyers’ demand largely approximated equilibrium predictions. However, at a more nuanced level, Period 1 demand tended to be higher (lower) than the rational expectations best responses at high (low) prices relative to the equilibrium level.
- In Condition I9 (high capacity constraint; low capacity):
- (a)
- Sellers were often able to price in period 1 at equilibrium level, which captured optimal ex-post profit on average;
- (b)
- As the session progressed, the sellers learned to increase their period 1 prices towards the equilibrium level, while the buyers generally became more strategically sophisticated vis a vis both myopic buying and irrational waiting deviations.

- In Condition I16 (relatively low capacity constraint; high capacity):
- (a)
- On average, the ex-post most profitable price in period 1 was higher than the equilibrium level. The sellers were boundedly strategic, as their prices often exhibited strategic upward adjustments from equilibrium to profit from the buyers with limited strategic sophistication, but also often exhibited a bias towards the overall equilibrium price;
- (b)
- From early on till the end of the session, the sellers set their period 1 prices around levels that were higher than equilibrium; learning among the buyers was not significant.

- Firms’ dynamic pricing strategies should not always be formulated as if the consumer market is highly strategic, nor as if the market is highly myopic. When there is a high capacity constraint, it is probably optimal or near-optimal to assume that the market is highly strategic. However, with lower capacity, assuming that the market is highly strategic (just like assuming that it is highly myopic) might lead to suboptimal profits. Firm decision makers should avoid falling into the trap of over- or underestimating the consumer market’s strategic sophistication.
- The consumer market may exhibit high (though potentially limited) strategic sophistication in correctly expecting potential future discounts, even if individual consumers might appear to be unsophisticated. At a more nuanced level, the market might adjust demand upwards less than it should when the price is low and adjust demand downwards less than it should when the price is high. This indicates that individual consumers could improve in savviness in terms of pre-empting potential future discounts.

#### Future Directions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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1 | In this statement and the text, we have implicitly adopted the tie-breaking rule that, if a consumer is indifferent between attempting to purchase now and not doing so, she chooses the former. This is simply motivated by expositional convenience: in period 1 in equilibrium only a consumer of a very specific valuation, namely v*, will face a tie-breaking problem; hence, the tie-breaking rule makes no impact on equilibrium characterizations given the assumption that the consumer population is large. |

2 | We also have analyzed the relationship between period 1 price and an “interim” profit variable that is definable for every round played in the experiment. Specifically, the interim profit in a round is equal to the sum of: (1) the ex-post period 1 profit in that round; and (2) if the round proceeded to period 2, the discounted period 2 profit given the remaining inventory at the beginning of that period if the seller priced optimally according to the gray line in Figure 4. Our conclusions regarding which period 1 price led to the optimal interim profit are similar to those regarding optimal ex-post profits for both conditions. |

3 | Statistical tests—using a GEE approach (see Section 5.4) to account for within subject/session dependencies—show that, in Condition I16, the ex-post round profit was higher than the overall equilibrium prediction at p < 0.01 when the period 1 price was 110 or 120, but not so (p > 0.1) when it was the overall equilibrium level of 100. |

**Figure 1.**Observed price (thick solid line ) and demand (gray vertical spike ) in period 1 by round and experimental session. Note that “demand” refers to attempted purchases which might not be all successful. Isolated outlier observations of prices above 200 units (five in each condition) have been excluded. The gray horizontal line ( ) and dashed line ( ) indicate, respectively, the equilibrium period 1 price and the optimal price for one-period selling. Each solid dot ( ) indicates the demand if all buyers followed the rational expectations equilibrium given the period 1 price, while each hollow diamond ( ) indicates the demand if all buyers were myopic. Where the two coincide, a solid diamond can be observed.

**Figure 2.**Ex-post mean discounted round profit (thick solid line ) and number of observations (bars ) by period 1 price. Gray line ( ) indicates overall equilibrium round profits and gray bar ( ) indicates the overall equilibrium price. In each panel, the range of prices on the horizontal axis captures more than 96% of the observations (303 and 304 out of 315 observations in Condition I9 and Condition I16, respectively); the ex-post mean discounted round profits with prices outside the range are all lower than with those inside the range.

**Figure 3.**Observed period 1 demands (solid dots ) by period 1 price and experimental condition. Isolated outlier observations of prices above 200 units (five in each condition) have been excluded. Also exhibited are the mean observed period 1 demand (thick solid line ), rational expectations equilibrium demand (gray line ), and myopic demand (dashed line ), by period 1 price.

**Figure 4.**Observed period 2 prices (solid dots ) by remaining inventory at the beginning of period 2 and experimental condition. Isolated outlier observations of prices above 200 units (two in Condition I9 and three in Condition I16) have been excluded. Also exhibited are the mean observed price (thick solid line ) and the benchmark optimal period 2 price (gray line ) by remaining inventory (see also Table A in Supplementary Online Appendix A). Note that, in equilibrium, the market should not have reached period 2 in Condition I9, and should have reached period 2 with four remaining units in Condition I16; also, in the Condition I16 sessions, no round had a period 2 with 14 units of remaining inventory, and hence there are no corresponding price observations.

Price in Period 1 (${\mathit{p}}_{1}*$) | Cutoff Valuation for Purchases in Period 1 (v*) | Price in Period 2 (${\mathit{p}}_{2}*$) | Season Demand (=Season Sales) | Season Profit (π*) | |
---|---|---|---|---|---|

$0<I\le {I}_{1}$ (One-period) | 1−I | 1−I | – | I | I(1−I) |

${I}_{1}<I\le {I}_{2}$ (Type I two-period) | $\frac{1+{\delta}_{F}-({\delta}_{F}+\delta )I}{2}$ | $\frac{1+{\delta}_{F}-2\delta -({\delta}_{F}-\delta )I}{2(1-\delta )}$ | 1−I | I | $H(I;\delta ,{\delta}_{F})$ |

${I}_{2}\le I\le 1$ (Type II two-period) | $\frac{{(2-\delta )}^{2}}{2(4-2\delta -{\delta}_{F})}$ | $\frac{2-\delta}{4-2\delta -{\delta}_{F}}$ | $\frac{2-\delta}{2(4-2\delta -{\delta}_{F})}$ | $\frac{6-3\delta -2{\delta}_{F}}{2(4-2\delta -{\delta}_{F})}$ | $\frac{{(2-\delta )}^{2}}{4(4-2\delta -{\delta}_{F})}$ |

**Table 2.**Rational expectations equilibria in the experiment given the period 1 price. Cutoff valuation (column 4) is the minimum valuation for buying to be a best response in period 1. Note that “mbd” = myopic buying demand in period 1 = max{0, min{20,(240-p

_{1})/10}}. The shaded rows indicate overall equilibria with the period 1 price that optimize total discounted round profit.

Period 1 | Period 2 (Given p_{1} and Equil. Purchases in Period 1) | Total Discounted Round Profit in Equil. Given p_{1} | |||||||
---|---|---|---|---|---|---|---|---|---|

Price (p_{1}) | Equil. Demand Given p_{1} | Equil. Period Profit Given p_{1} | Cutoff Valuation for Purchases (v**) Given p_{1} | No. of Buyers with Valuation > p_{1} However, Hold off Purchase | Remaining Inventory at Beginning | Equil. Price | Equil. Demand | Undiscounted Equil. Period Profit | |

Condition I9 | |||||||||

<150 | mbd | 9p_{1} | p_{1} + 5 | 0 | No period 2 | NA | NA | NA | 9p_{1} |

150 | 9 | 1350 | 155 | 0 | No period 2 | NA | NA | NA | 1350 |

160 | 7 | 1120 | 175 | 1 | 2 | 150 | 2 | 300 | 1270 |

170 | 5 | 850 | 195 | 2 | 4 | 150 | 4 | 600 | 1150 |

180 | 3 | 540 | 215 | 3 | 6 | 150 | 6 | 900 | 990 |

190 | 1 | 190 | 235 | 4 | 8 | 150 | 8 | 1200 | 790 |

≥200 | 0 | 0 | NA | mbd | 9 | 150 | 9 | 1350 | 675 |

Condition I16 | |||||||||

<80 | mbd | 16p_{1} | p_{1} + 5 | 0 | No period 2 | NA | NA | NA | 16p_{1} |

80 | 16 | 1280 | 85 | 0 | No period 2 | NA | NA | NA | 1280 |

90 | 14 | 1260 | 105 | 1 | 2 | 80 | 2 | 160 | 1340 |

100 | 12 | 1200 | 125 | 2 | 4 | 80 | 4 | 320 | 1360 |

110 | 10 | 1100 | 145 | 3 | 6 | 80 | 6 | 480 | 1340 |

120 | 8 | 960 | 165 | 4 | 8 | 80 | 8 | 640 | 1280 |

130 | 7 | 910 | 175 | 4 | 9 | 80 | 9 | 720 | 1270 |

140 | 6 | 840 | 185 | 4 | 10 | 90 | 9 | 810 | 1245 |

150 | 4 | 600 | 205 | 5 | 12 | 100 | 10 | 1000 | 1100 |

160 | 3 | 480 | 215 | 5 | 13 | 100 | 11 | 1100 | 1030 |

170 | 2 | 340 | 225 | 5 | 14 | 110 | 11 | 1210 | 945 |

≥180 | 0 | 0 | NA | mbd | 16 | 120 | 12 | 1440 | 720 |

**Table 3.**Observed per round means of major period 1 dependent variables by condition and block, with (s.d. in parentheses). In columns 2 and 3, where a mean is significantly different from the overall equilibrium (see the gray rows in Table 2) according to a t test, it is marked by one or more asterisks (* p ≤ 0.05, ** p < 0.01). Column 5 pertains to deviations comparing buyers’ decisions with rational expectations best responses given the period 1 price; where a mean in that column is significantly different from zero according to a t test, it is marked according to similar conventions as in columns 2 and 3.

Price (p_{1}) | Period Profit | Demand | Deviation of Demand from Best Response | |
---|---|---|---|---|

Condition I9 | ||||

Overall equilibrium prediction | 150 | 1350 | 9 | 0 |

Block 1 | 137.43 (7.39) * | 1065.71 (38.83) ** | 9.71 (0.71) | −0.60 (0.35) * |

Block 2 | 149.43 (10.46) | 1126.38 (92.11) ** | 8.37 (1.33) | −0.04 (0.50) |

Block 3 | 154.38 (8.37) | 1094.48 (36.80) ** | 7.91 (0.90) | −0.08 (0.24) |

Condition I16 | ||||

Overall equilibrium prediction | 100 | 1200 | 12 | 0 |

Block 1 | 106.19 (10.20) | 1131.90 (34.04) * | 11.41 (0.92) | 0.55 (0.75) |

Block 2 | 107.05 (2.75) ** | 1177.71 (63.05) | 11.66 (0.67) | 0.63 (0.53) |

Block 3 | 111.71 (7.70) ** | 1160.67 (33.42) | 11.15 (0.69) | 0.82 (0.26) ** |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Mak, V.; Rapoport, A.; Gisches, E.J. Dynamic Pricing Decisions and Seller-Buyer Interactions under Capacity Constraints. *Games* **2018**, *9*, 10.
https://doi.org/10.3390/g9010010

**AMA Style**

Mak V, Rapoport A, Gisches EJ. Dynamic Pricing Decisions and Seller-Buyer Interactions under Capacity Constraints. *Games*. 2018; 9(1):10.
https://doi.org/10.3390/g9010010

**Chicago/Turabian Style**

Mak, Vincent, Amnon Rapoport, and Eyran J. Gisches. 2018. "Dynamic Pricing Decisions and Seller-Buyer Interactions under Capacity Constraints" *Games* 9, no. 1: 10.
https://doi.org/10.3390/g9010010