An analytical game perspective model for pay-what-you-want pricing schemes considering consumer fairness

The pay-what-you-want (PWYW) pricing scheme allows consumers to pay whatever amount they wish for a particular product or service. PWYW is increasingly being used by restaurants and the hotel industry in Western countries, and it is thus important to study consumer behavior when faced with the PWYW option and to further explore the acceptance of PWYW conditions from a theoretical perspective. The extant literature indicates that the perception a “fair price” in the mind of a consumer is the result of an evaluation based on comparing the price paid with the reference price. Fairness thus plays an important role in affecting consumers’ intentions to pay more than zero under the PWYW option. First, we develop a theoretical model that incorporates the consumer’s “fair price” into their utility function. Thus, we obtain different decisions and optimal Nash equilibrium prices across individuals. When a seller provides differentiated products and allows consumers the option to pay what they want, it is fascinating that, in our model setting, the scenario in which all consumers pay zero never provides an equilibrium solution as long as the consumers’ fair prices are positive. In contrast, a scenario in which all consumers are willing to pay a nonzero amount occurs under certain conditions. Furthermore, it is demonstrated that it is possible for this pricing system to be more profitable for the seller than a uniform pricing scheme. Finally, we conduct a sensitivity analysis of the parameters in our proposed model and present several illustrative examples to verify our results.

1. When \(p_i\geqslant r_i\), the relative utility \(U(r_i,p_i)=U_i^{\textrm{PWYW}}-U^B_i=(r_i-p_i)(1+\vartheta )<0\), so this case does not need to be considered.

2. This has the same derivation as \(\bar{r}_i(r_j)\); thus, \(\bar{r}_j(r_i)\) is given by

   $$\begin{aligned} r_j>\left( \frac{1}{B(r_i)}-\frac{1}{(1-\theta )}\right) \frac{4\lambda B^2(r_i)}{(1-2\theta )}=\bar{r}_j(r_i), \end{aligned}$$

   where

   $$\begin{aligned} B(r_i)=\left[ 1-\frac{1}{(1-2\theta )} \left( 1-\sqrt{1-\left( \frac{1-2\theta }{1-\theta }\right) \frac{r_i}{\lambda }}\right) \right] . \end{aligned}$$

3. Consumer j’s mixed strategy is symmetric to consumer i’s strategy, and by symmetry we can write

   $$\begin{aligned} \tau _j=\frac{W(\gamma _j,0)-(P_r+\gamma _i-\alpha )}{W^{\textrm{FR}}(\gamma _i,\gamma _j)-W (\gamma _i,\gamma _j)+W (\gamma _i,0)- (P_r+\gamma _i-\alpha )} \end{aligned}$$

The authors gratefully acknowledge financial support from the General Project of Philosophy and Social Science Research in Colleges and Universities in Jiangsu Province (Grant No. 2020SJA0174), the Natural Science Foundation of Jiangsu Higher Education Institution of China (Grant No. 21KJB410001), the National Science Foundation of China (Grant No. 71871121), the Startup Foundation for Introducing Talent of NUIST (Grant No. 1441182001002), and the Future Network Scientific Research Fund Project (Grant No. FNSRFP-2021-YB-19). The authors are most grateful to the anonymous referees and the editor for their valuable comments and suggestions that greatly improve this paper both in contents and representations.

Authors and Affiliations

1. School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing, 210044, Jiangsu, China

   Guang Yang, Mulin Liu & Mei Cai

2. School of Business, Shaoxing University, Shaoxing, 312000, Zhejiang, China

   Qihua Yin

Corresponding author

Correspondence to Guang Yang.

Conflict of interests

The authors declare that they have no conflicts of interest.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Proofs of propositions

1.1 A.1 Proof of Proposition 3

Proof

When a consumer free-rides, \(p_i = 0\), she or he incurs the highest social cost \(\alpha \). Consumer i prefers PWYW pricing when \(U_i^{\textrm{PWYW}}-U_i> 0\) or the consumer’s relative utility is given by

where the single-period consumer surplus of free-riding is \(U(P_r,0)=P_r-t(1-2x_i)-\alpha \left[ \frac{P_r-p_i}{P_r}\right] .\)

As described above, consumer behavior is determined by a dynamic model based on contributions by the consumers together with the consideration of the firm’s survival. The following two-period Bellman equation summarizes the dynamic model:

Two equilibria when consumers contribute are derived. The first equilibrium is the free-rider outcome. In this case, consumer i’s contribution is set to zero, \(p_i=0\). The value function reduces to a geometric series having the well-known solution

where \(W^{\textrm{FR}}(P_r,0,p_j)\) increases with both the contribution by consumer j and the discount factor, but it decreases with increasing social cost. In the absence of contributions made by consumer j, the dynamic model collapses to a one-shot static outcome of \(U(P_r,0)\) and the PWYW firm fails. The second equilibrium is an interior solution to the consumers’ objective functions. The first-order condition with respect to the consumers’ levels of contribution is

where the marginal cost of contributing, \(1-\frac{\alpha }{P_r}\), decreases with the social cost, \(\alpha \), but it increases with respect to the reference price. The marginal benefit of contributing to the survival of the firm is \(\frac{\beta W(P_r^{'},p_i^{'},p_j^{'})}{2\bar{F}}\left( \frac{p_i+p_j}{\bar{F}}\right) ^{-\frac{1}{2}}\). The optimal contribution level and value function are solved using similar methods as those previously described in the basic model. The consumer value function is

where \(\lambda =\bar{F}/\beta ^2\) and \(\gamma _i=-t(1-2x_i)\). The value function including fairness considerations reduces to the baseline case when \(\alpha = 0\) and \(\gamma _i\) is interpreted in the same way as \(v_i\) in the previous model. Consumer i’s contribution level can be expressed as a function of consumer j’s contributions by substituting the optimal value function \(W(P_r,p_i,p_j)\) into Eq. (30). The reaction function in contribution levels is given by

where contributions increase strictly with the reference price, \(\partial p_i/\partial P_r>0\), and consumer i’s relative value, \(\gamma _i\). The strategic interaction between consumers is dependent on their levels of contribution. The marginal effect of an increase in the contribution made by consumer j on consumer i’s contribution is

which is positive (strategic complements) when \(p_j > \frac{3\lambda }{4}-\frac{P_r(\gamma _i-\alpha )}{P_r-\alpha }\) and negative otherwise (strategic substitutes). Contributions made by consumer j above this threshold provide an incentive to consumer i to contribute the necessary funds to ensure the firm’s survival. A marginal increase in the social cost of fairness increases the optimal contribution if a consumer’s value for the good is higher than the reference price, \( \gamma _i> P_r\), otherwise contributions decrease as the social cost increases. If consumer j does not contribute, then consumer i’s optimal contribution is

and consumer j’s free-rider utility is

The reaction function for consumer j is symmetric to that of consumer i. The Nash equilibrium in contributions is the point of intersection of the two reaction functions. At this point, the optimal contribution level for consumer i is

and the difference in contribution levels between consumers is proportional to each consumer’s value for the good, \(p_i- p_j = \frac{P_r(\gamma _i-\gamma _j)}{P_r-\alpha }\). The lifetime utility of consumer i when both consumers contribute is found by replacing \(p_j\) in Eq. (33) with the Nash equilibrium contributions

The normal-form game in the presence of “fairness” is summarized in Table 7.

The equilibrium outcome is dependent on the relative difference in consumer valuations, \(\triangle \gamma =\gamma _i-\gamma _j\), the social cost parameter \(\alpha \), and the reference price \(P_r\). \(\square \)

1.2 A.2 Proof of Proposition 4

Proof

We assume that the fair price for consumers follows a uniform distribution on the interval [0, r]. For a risk-neutral firm, the optimal uniform price per consumer is \(\frac{r}{2}\) and the total expected profit is \(\pi ^u=\frac{r}{2}-F\), where F denotes the fixed costs. The firm receives the minimum revenue under the PWYW option, expressed by:

Proposition 1 indicates that the case in which all consumers are free-riders can never attain an equilibrium, and the amount consumer i is willing to pay is not more than F when his or her fair price \(r_i>\frac{\bar{F}(2-\beta )}{\beta }\). Thus, a positive profit for the firm is guaranteed when \(r_i > 0\), and this profit can be expressed as:

From the two profit functions in the different pricing schemes, it is easy to derive the condition under which PWYW is more profitable than fixed pricing (\(\pi ^{\mathrm {\textrm{PWYW}}}>\pi ^u\)). We first consider the case in which a consumer’s fair price for the product is more than \(\tilde{r}\), i.e., \(r> r_i>\tilde{r} > \frac{r}{2}\). It is confirmed that PWYW is not only more profitable, but also that the firm makes at least normal profits in this case, that is, \(\pi ^{\textrm{PWYW}}=\bar{F}-F\). Next, we consider the case in which a single consumer cannot guarantee a positive profit for the firm, \(r>\tilde{r }>r/2\). This case leads to the firm having greater expected profits using the PWYW pricing scheme, \(\pi ^{\textrm{PWYW}}>\pi ^{u}\), such that:

or

The probability of this event is:

which is greater than zero when \(r\geqslant \frac{8\lambda (1-\theta )^2}{(2\theta ^2-3\theta +3)^2}\). In fact, this lower bound guarantees that only one consumer is willing to contribute. When all consumers are willing to contribute, the probability increases from that in Eq. (44). \(\square \)

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