Analysis of Fairness and Incentives of Profit Sharing Schemes in Group Buying

Abstract

Payoff distribution within coalitions in group-buying environments, where a group of buyers pool their demands to benefit from volume discounts, is a well-studied problem. However, the general assumption in literature is unit demand, where every buyer needs one item. In the case of varying volume demands, both the valuation and the contribution of buyers will change. In this paper, we introduce the variable demand group-buying game with implied values, where the valuation of one item for the buyer is equal to the unit price, which the buyer can obtain by itself. Buyers with higher volumes of demand have lower valuation per unit. We consider scenarios where volume discounts kick in at multiple volume thresholds and investigate the effect of different profit sharing mechanisms in coalitions of buyers: proportional cost sharing based on volume demand and valuation, proportional profit sharing based on volume and contribution, and adjusted Clarke mechanism. All these mechanisms are efficient, budget-balanced, and individual rational. We evaluated these five payoff mechanisms on the following criteria: stability, incentive compatibility, and fairness. We introduce a fairness criteria that correlates with marginal contribution. Experimental results show that fairness and stability are difficult to satisfy simultaneously.

Appendix

Proof 1

Variable demand group-buying game is a non-convex game: A game is convex when $\forall _{S,T}\;u(C_{S \cup T}) \ge u(C_S) + u(C_T) - u(C_{S \cap T})$ is satisfied. We will prove that variable demand group-buying game is not convex by giving a counter example. Let’s say $S=\{b_1,b_2,b_3,b_4\}$ and $T=\{b_1,b_2,b_3,b_5,b_6\}$ are two sets of buyers and every $b_i$ has a demand of 1. $S\cap T=\{b_1,b_2,b_3\}$ and $S\cup T=\{b_1,b_2,b_3,b_4,b_5,b_6\}$.

$$\begin{aligned} p(n) = \left\{ \begin{array}{l l} 10 &{} \quad n < 4\\ 9 &{} \quad n = 4\\ 8 &{} \quad n > 4\\ \end{array} \right. \end{aligned}$$

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Given the price schedule $p(n)$, the utilities become: $u(C_S)=4$, $u(C_T)=10$, $u(C_{S \cup T})=12$, and $u(C_{S \cap T})=0$. When we set these values into the convex game criteria $12 \ge 4+10-0$, the condition is not satisfied. Hence, variable demand group-buying game is a non-convex game.

Proof 2

Cost sharing proportional to volume mechanism is in the core:

$$\begin{aligned} \forall S\subseteq N, \sum _{i \in C_S} u_i(C_N) \ge u(C_S) \nonumber \\ \sum _{i \in C_S} \big [p(n_i)-p(n_T)\big ] \ge \sum _{i \in C_S} \big [p(n_i)-p(n_S)\big ] \nonumber \\ p(n_S) \ge p(n_T)\nonumber \\ \end{aligned}$$

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