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Bargaining and inequity aversion: On the efficiency of the double auction
Economics Letters 114 (2012) 178–181
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Bargaining and inequity aversion: On the efficiency of the double auction✩ Alexander Rasch, Achim Wambach ∗ , Kristina Wiener University of Cologne, Germany
article
info
Article history: Received 24 November 2010 Received in revised form 13 September 2011 Accepted 23 September 2011 Available online 8 October 2011
abstract In a bargaining setting with asymmetrically informed, inequity-averse parties, a fully efficient mechanism (i.e., the double auction) exists if and only if compassion is strong. Less compassionate parties do not trade in the double auction in the limit of strong envy. © 2011 Elsevier B.V. All rights reserved.
JEL classification: C78 D44 D82 Keywords: Bargaining Double auction Inequity aversion Mechanism design Two-sided asymmetric information
1. Introduction In this paper, we are interested in whether behavioral economics can provide new insights into the strategic behavior in bargaining situations with two-sided incomplete information. To this end, we investigate what happens if participants are inequity averse à la Fehr and Schmidt (1999)1 as inequity aversion seems to play an important role in one-to-one bargaining situations. In their setup, the utility of a person decreases whenever the final split is unfair. Their model of inequity aversion is made up of two parts: an envy as well as a compassion factor. We show that only if compassion is sufficiently important, there exists a mechanism which ensures that all possible gains from trade are realized and both parties do not suffer any disutility from inequity. In this case, the double auction implements full efficiency. The double auction was first analyzed by Chatterjee and Samuelson (1983). In a double
✩ We would like to thank Andreas R. Engel, Florian Englmaier, Vitali Gretschko, Jesko Herre, and Wanda Mimra for very helpful comments and discussions. We also gratefully acknowledge valuable suggestions by an anonymous referee. We are indebted to Matthias Schott for his help with the numerical analysis. ∗ Correspondence to: University of Cologne, Albertus-Magnus-Platz, 50923 Cologne, Germany. Tel.: +49 0 221 470 5822; fax: +49 0 221 470 5024. E-mail address: [email protected] (A. Wambach). 1 For an alternative approach to modeling fair-minded behavior, see Bolton and
Ockenfels (2000). Linhart (2001) assumes both seller and buyer to either minimize maximum regret or maximize maximum profit. 0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.09.027
auction, a seller and a buyer independently submit a bid. Trade only occurs if the price submitted by the seller is smaller than the buyer’s bid.2 In the case without inequity aversion, inefficiency arises because in order to get a better deal, the seller tends to overstate her costs and the buyer has an incentive to understate his valuation. With sufficiently strong compassion, such a behavior would result in disutility as any additional gain is necessarily accompanied by a loss to the other party. Having a closer look at the efficiency of the double auction if compassion plays a less crucial role, we show that in the limit of strong envy, trade breaks down. If the envy factor is finite, we analyze the equilibrium in the double auction numerically. We find that an increase in the envy factor leads to bidding strategies which are further away from truth-telling. The opposite is true for an increase in the compassion factor. As a result, the lower (higher) the degree of envy (compassion), the more efficient the bargaining outcome. There exists some theoretical work on bargaining where inequity aversion is considered. These articles differ from the present work with respect to both the bargaining setup and the informational assumptions (Fehr and Schmidt, 1999; Lopomo and Ok, 2001; Ellingsen and Johannesson, 2004; Ewerhart, 2006; Montero, 2007; von Siemens, 2009).
2 Myerson and Satterthwaite (1983) show for uniformly distributed costs and valuations that the double auction is the most efficient mechanism which, however, is ex post inefficient.
A. Rasch et al. / Economics Letters 114 (2012) 178–181
The present article contributes to the literature that addresses the question to what extent inequity aversion mitigates or aggravates problems resulting from informational asymmetries. In some cases, efficiency decreases due to inequity aversion (see, eg., Grund and Sliwka, 2005; Bartling and von Siemens, 2010b; Englmaier and Wambach, 2010) whereas informational problems may also become less severe if inequity aversion is prevalent (Kragl and Schmid, 2009; Bartling and von Siemens, 2010a; Küçükşenel, 2010). Our paper is in particular related to the small part of the literature which explicitly distinguishes between the potentially different effects of envy and compassion (Grund and Sliwka, 2005; Montero, 2007; Rey-Biel, 2008). 2. Model and results We analyze a bargaining problem where a seller produces a good a buyer wants to buy. Denote by c the costs the seller has to incur when producing the good and by v the value of the object for the buyer. Both costs and valuations are assumed as independent random variables which are distributed over a given interval c ∈ [c , c¯ ] and v ∈ [v, v¯ ] (with c < v¯ ). f (v) and g (c ) are the respective probability density functions which are assumed continuous and positive on their domains. The corresponding distribution functions are F (c ) and G(v). Costs and valuations are private information and hence only known to the respective party. Consider a mechanism ζ = (x(ˆc , vˆ ), p(ˆc , vˆ )) where x(ˆc , vˆ ) ∈ [0, 1] is the allocation rule and p(ˆc , vˆ ) is the payment rule conditional on trade taking place. Both rules depend on the messages cˆ and vˆ sent by the seller and buyer, respectively. Suppose the other party tells the truth, then following Fehr and Schmidt (1999), the expected utility from bargaining is given by E[US (c , cˆ )] =
v¯
∫ v
x(ˆc , v)(p(ˆc , v) − c
− α max{0, −(p(ˆc , v) − c ) + (v − p(ˆc , v))} − β max{0, (p(ˆc , v) − c ) − (v − p(ˆc , v))})g (v)dv and E[UB (v, vˆ )] =
c¯
∫
x(c , vˆ )(v − p(c , vˆ ) c
− α max{0, −(v − p(c , vˆ )) + (p(c , vˆ ) − c )} − β max{0, (v − p(c , vˆ )) − (p(c , vˆ ) − c )})g (c )dc where subscripts S and B denote the seller and buyer, respectively. If trade takes place, the first two terms in the brackets represent the gross gains of trade to the seller and the buyer. The third and the fourth terms reflect the disutility the parties incur from an unfair division of the surplus where the third term gives the disadvantage weighted with the envy factor α . The fourth term accounts for the advantage to the seller or buyer weighted with the compassion factor β . It is assumed that α ≥ β and 0 ≤ β ≤ 1.3 Due to the revelation principle, we restrict our attention to incentive-compatible direct mechanisms (Myerson, 1981). The maximization problem can be written as a weighted sum of the individual expected utilities
179
as well as the individual-rationality constraints E[US (c , c )] ≥ 0, E[UB (v, v)] ≥ 0. In what follows, we will look for a fully efficient mechanism that can be implemented when agents are inequity averse. Note that (ex post) efficiency refers to two aspects: first, the allocation is efficient whenever all possible gains from trade are realized, i.e., trade occurs for c ≤ v . Second, the payment rule must be such that agents do not incur disutility from inequity aversion, i.e., the gains from trade are shared equally. Hence, the optimal allocation and payment rules are given by x∗ (c , v) =
1 0
if c ≤ v else
(1)
.
(2)
and p∗ (c , v) =
c+v 2
As the following proposition shows, full efficiency can be implemented for general distributions of costs and valuations if and only if compassion is strong enough. Proposition 1. The mechanism ζ ∗ = (x∗ (ˆc , vˆ ), p∗ (ˆc , vˆ )) is incentive compatible and thus implements full efficiency if and only if β ≥ 1/2. Proof. See the Appendix.
The intuition behind this result is as follows. Consider the seller side: if a seller expecting the buyer to send a message equal to his valuation reveals her true costs, the optimal payment lies halfway between costs and valuation if trade occurs. Under these circumstances, both parties are equally well off for any value of both costs and valuation and there is no disutility from inequity on either side. If, on the other hand, the seller’s message is above her actual costs, this leads to a better payment from her point of view. At the same time, however, this results in an unequal final bargaining outcome. Now if compassion is of sufficiently great importance, the seller prefers to tell the truth concerning her costs. Next consider the double auction as a possible mechanism. In the double auction, both the seller and the buyer submit bids. Trade occurs whenever the seller’s bid bS is equal to or smaller than the buyer’s bid bB at a price that is equal to the average bid, i.e., p = (bS +bB )/2. Hence, if the seller and the buyer submit bids equal to their true costs and valuation, respectively, the allocation and the payment are identical to expressions (1) and (2) of the optimal mechanism. Therefore, the next result follows immediately from Proposition 1. Corollary 1. The double auction implements full efficiency if β ≥ 1/2.4 , 5 We use the double auction to address the case where compassion is weak. As the following proposition points out, the situation becomes very much different if envy is strong. Proposition 2. Suppose that v ≤ c and/or v¯ ≤ c¯ and that β < 1/2. Then, in the limit where α → ∞, trade occurs with probability zero in equilibrium in the double auction.
max λE[US (c , c )] + (1 − λ)E[UB (v, v)] x ,p
subject to the incentive-compatibility constraints E[US (c , c )] ≥ E[US (c , cˆ )], E[UB (v, v)] ≥ E[UB (v, vˆ )]
3 For a discussion of these assumptions, see Fehr and Schmidt (1999).
4 In their setup, Chatterjee and Samuelson (1983) show that truth-telling is also an equilibrium if both sides are very (infinitely) risk averse. 5 Note that truth-telling with its fair outcome is not the only strategy which leads to a fair allocation with trade in the double auction. Such an allocation can also be obtained if both sides bid according to a strategy with slope 1, i.e., bS (c ) = c + ∆ and bB (v) = v − ∆ with ∆ ≶ 0. See Rasch et al. (2011) for details.
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A. Rasch et al. / Economics Letters 114 (2012) 178–181
Fig. 1. Equilibrium bidding strategies for the seller for different values of α and β : (a) α = β = 0 (dashed line), (b) β ≥ 1/2 (dashed line), (c) α = 1/2, β = 0.495, (d) α = 1/4, β = 0, and (e) α = 10, β = 0.
Fig. 2. Trade regions (upper left corner) for different values of α and β : (a) α = β = 0 (dashed line), (b) β ≥ 1/2 (dashed line), (c) α = 1/2, β = 0.495, (d) α = 1/4, β = 0, and (e) α = 10, β = 0.
Proof. See the Appendix.
omitted. Consider the seller who deviates from the truth-telling strategy and sends a message cˆ > c. Assume that the buyer sticks to the truth-telling strategy, i.e., vˆ = v . Hence, given expressions (1) and (2), the seller’s expected utility is given by
If compassion is weak, trade no longer takes place as envy becomes extremely important. This is rather intuitive. If an equal split is not possible, then one trading partner will be worse off than the other. Now as the importance of envy increases, the less attractive will be the trade outcome for the side that is worse off. The only ‘fair’ outcome which is then reachable is the no-trade outcome.6 Next consider the case where β < 1/2 and α is finite. As it turns out, a general solution to the problem is not analytically tractable. Therefore, we resort to numerical simulations. Figs. 1 and 2 illustrate the seller’s equilibrium bidding strategies in the double auction7 for different parameter values of inequity aversion as well as the resulting trade region (for uniformly distributed costs and valuations). The buyer’s strategies are analogous. If the costs for supplying the product are high, the seller bids more aggressively (cases (d) and (e)) than without inequity aversion (case (a)) due to envy. Now if envy becomes more important, the bid will be even higher (compare case (d) to case (e)). On the other hand, if the costs are low, obtaining more trade through a lower bid is optimal compared to the situation without inequity aversion (cases (c) and (d)). As the importance of compassion rises (case (c)), the seller bids less aggressively in order to avoid being better off. The numerical analysis shows that an increase in the degree of envy leads to less trade (compare cases (d) and (e)). This is in line with Proposition 2 where, in the limit, trade breaks down completely. On the other hand, an increase in the degree of compassion results in more trade (compare cases (c) and (d)). Moreover, as shown in Corollary 1, if β converges to 1/2, truthtelling becomes optimal and trade occurs whenever it is efficient (see case (b)). Appendix A.1. Proof of Proposition 1
E[US (c , cˆ )] =
v¯
∫
cˆ + v 2
cˆ
∫ v¯
−β
=
cˆ + v 2
cˆ
=
1
∫
− c g (v)dv cˆ + v 2
cˆ
v¯
∫
cˆ + v −c − v− g (v)dv
− c − β cˆ − c
2
g (v)dv
v¯
v g (v)dv 1 + cˆ − β − c (1 − β) (1 − G(ˆc )). 2 cˆ
2
Note that if trade occurs, this will lead to a price greater than (c + v)/2, i.e., only the advantage term is relevant for the seller. Trade occurs whenever v ≥ cˆ . It follows that
∂ E[US (c , cˆ )] = ∂ cˆ
1 2
− β (1 − G(ˆc )) − (ˆc − c )(1 − β)g (ˆc ).
For truth-telling to be optimal, ∂ E[US (c , cˆ )]/∂ cˆ < 0 ∀ˆc > c must hold. Since 1 − G(ˆc ) ≥ 0, this is always true whenever β ≥ 1/2. On the other hand, if β < 1/2, then for cˆ = c, it follows that
∂ E[US (c , cˆ )] ∂ cˆ
> 0, cˆ =c
i.e., the mechanism is not incentive compatible. Now consider the case where cˆ < c. Under the rule that gains from trade are split equally, such a message not only leads to a reduction in the seller’s payoff by passing more to the buyer but also yields additional disutility by favoring the buyer. Therefore, a message cˆ < c is never optimal.
Proof. We prove the proposition for the seller only. The case for the buyer can be analyzed in an analogous fashion and is therefore
A.2. Proof of Proposition 2
6 If v > c, equilibria where b (c ) = c + ∆ and b (v) = v − ∆ with ∆ > 0 exist S B in the double auction even for β < 1/2 (Rasch et al., 2011). 7 Note that Fig. 1 only shows the equilibrium bidding behavior for those seller
Proof. We prove the proposition by contradiction. Assume that there exists an equilibrium where trade occurs with strictly positive probability for any value of α . From Proposition 1 above and Proposition 2 in Rasch et al. (2011), we know that fair allocations are not an equilibrium under the assumptions of Proposition 2.
types who trade. Seller types who do not trade may submit any bid as long as bS (c ) > bB (v) ∀v holds (e.g., bS (c ) = 1).
A. Rasch et al. / Economics Letters 114 (2012) 178–181
Denote the sets of types who trade by HS and HB . A seller with costs c ∈ HS who trades with buyers of types v ∈ HB may be worse off, equally well off, or better off compared to her trading partner. We thus define the following sets which correspond to these cases: AS (c , α) := {v ∈ HB |(c + v)/2 > p(c , v, α)}, BS (c , α) := {v ∈ HB |(c + v)/2 = p(c , v, α)}, and CS (c , α) := {v ∈ HB |(c + v)/2 < p(c , v, α)} where p(c , v, α) is the bargaining price that depends on α . The seller then gets an expected utility of E[US (c , v)] = E[US (c , v)|v ∈ AS (c , α)]
+ E[US (c , v)|v ∈ BS (c , α)] + E[US (c , v)|v ∈ CS (c , α)]. Consider first the case where limα→∞ Pr(AS (c , α)) = 0 ∀c ∈ HS and limα→∞ Pr(AB (v, α)) = 0 ∀v ∈ HB which implies that types who trade must bid their costs or valuations (±∆ with ∆ > 0), respectively. This, however, cannot be an equilibrium as shown in Proposition 2 in Rasch et al. (2011). Assume therefore that limα→∞ Pr(AS (c , α)) ≥ kS with kS > 0. The expected utility for a seller who trades amounts to E[US (c , v)] =
∫ AS (c ,α)
−α
(p(c , v, α))g (v)dv
∫ AS (c ,α)
(c + v − 2p(c , v, α))g (v)dv
+ E[US (c , v)|v ∈ BS (c , α)] + E[US (c , v)|v ∈ CS (c , α)]. Given that Pr(AS (c , α)) ≥ kS , it then holds that α → ∞ ⇒ E[US (c , v)] < 0 if and only if A (c ,α) (c + v − 2p(c , v, α))g (v)dv S > 0 for α → ∞. Assume that the opposite holds, i.e.,
∫ AS (c ,α)
(c + v − 2p(c , v, α))g (v)dv = 0 for α → ∞.
Let A′S (c , α) := {v ∈ AS (c , α)|c + v − 2p(c , v, α) > ϵ} ⊂ AS (c , α). We can thus conclude that
∫ lim
α→∞ A′ (c ,α) S
(c + v − 2p(c , v, α))g (v)dv = 0 ∀ϵ > 0
and hence lim Pr(A′S (c , α)) = 0 ∀ϵ > 0.
α→∞
Then, assuming that ‖ · ‖∞ is the supremum norm for set AS (c , α), it must hold that lim ‖c + v − 2p(c , v, α)‖∞ = 0
α→∞
a.s.
181
Hence, it is true that ∀ϵ > 0 ∃α˜ such that ∀α > α˜ ,
‖c + v − 2p(c , v, α)‖∞ < ϵ a.s. which is equal to sup
v∈AS (c ,α)
|c + v − 2p(c , v, α)| < ϵ a.s.
It thus follows that ∀ϵ > 0 ∃α˜ such that ∀α > α˜ and for almost all v ∈ AS (c , α), it must hold that
|c + v − 2p(c , v, α)| < ϵ. Then, however, we have limα→∞ p(c , v, α) = (c + v)/2 for almost all v ∈ AS (c , α). Therefore, we get limα→∞ Pr(AS (c , α)) = 0 which contradicts our initial assumption that Pr(AS (c , α)) ≥ kS ∀α ≥ 0. An analogous argument holds on the buyer side. References Bartling, B., von Siemens, F.A., 2010b. The intensity of incentives in firms and markets: Moral hazard with envious agents. Labour Economics 17, 598–607. Bartling, B., von Siemens, F.A., 2010a. Equal sharing rules in partnerships. Journal of Institutional and Theoretical Economics 166, 299–320. Bolton, G.E., Ockenfels, A., 2000. ERC: A theory of equity, reciprocity, and competition. American Economic Review 90, 166–193. Chatterjee, K., Samuelson, W., 1983. Bargaining under incomplete information. Operations Research 31, 835–851. Ellingsen, T., Johannesson, M., 2004. Promises, threats and fairness. Economic Journal 114, 397–420. Englmaier, F., Wambach, A., 2010. Optimal incentive contracts under inequity aversion. Games and Economic Behavior 69, 312–328. Ewerhart, C., 2006. The effect of sunk costs on the outcome of alternating-offers bargaining between inequity-averse agents. Schmalenbach Business Review 58, 184–203. Fehr, E., Schmidt, K.M., 1999. A theory of fairness, competition, and cooperation. Quarterly Journal of Economics 114, 817–868. Grund, C., Sliwka, D., 2005. Envy and compassion in tournaments. Journal of Economics & Management Strategy 14, 187–207. Küçükşenel, S., 2010. Behavioral mechanism design. Journal of Public Economic Theory (in press). Kragl, J., Schmid, J., 2009. The impact of envy on relational employment contracts. Journal of Economic Behavior & Organization 72, 766–779. Linhart, P.B., 2001. Bargaining solutions with non-standard objectives. Review of Economic Design 6, 225–239. Lopomo, G., Ok, E.A., 2001. Bargaining, interdependence, and the rationality of fair division. RAND Journal of Economics 32, 263–283. Montero, M., 2007. Inequity aversion may increase inequity. Economic Journal 117, 192–204. Myerson, R.B., 1981. Optimal auction design. Mathematics of Operations Research 6, 58–73. Myerson, R.B., Satterthwaite, M.A., 1983. Efficient mechanisms for bilateral trade. Journal of Economic Theory 29, 265–281. Rasch, A., Wambach, A., Wiener, K., 2011. The Double Auction with Inequity Aversion. Mimeo. Rey-Biel, P., 2008. Inequity aversion and team incentives. Scandinavian Journal of Economics 110, 297–320. von Siemens, F.A., 2009. Bargaining under incomplete information, fairness, and the hold-up problem. Journal of Economic Behavior & Organization 71, 486–494.
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Bargaining and inequity aversion: On the efficiency of the double auction✩ Alexander Rasch, Achim Wambach ∗ , Kristina Wiener University of Cologne, Germany
article
info
Article history: Received 24 November 2010 Received in revised form 13 September 2011 Accepted 23 September 2011 Available online 8 October 2011
abstract In a bargaining setting with asymmetrically informed, inequity-averse parties, a fully efficient mechanism (i.e., the double auction) exists if and only if compassion is strong. Less compassionate parties do not trade in the double auction in the limit of strong envy. © 2011 Elsevier B.V. All rights reserved.
JEL classification: C78 D44 D82 Keywords: Bargaining Double auction Inequity aversion Mechanism design Two-sided asymmetric information
1. Introduction In this paper, we are interested in whether behavioral economics can provide new insights into the strategic behavior in bargaining situations with two-sided incomplete information. To this end, we investigate what happens if participants are inequity averse à la Fehr and Schmidt (1999)1 as inequity aversion seems to play an important role in one-to-one bargaining situations. In their setup, the utility of a person decreases whenever the final split is unfair. Their model of inequity aversion is made up of two parts: an envy as well as a compassion factor. We show that only if compassion is sufficiently important, there exists a mechanism which ensures that all possible gains from trade are realized and both parties do not suffer any disutility from inequity. In this case, the double auction implements full efficiency. The double auction was first analyzed by Chatterjee and Samuelson (1983). In a double
✩ We would like to thank Andreas R. Engel, Florian Englmaier, Vitali Gretschko, Jesko Herre, and Wanda Mimra for very helpful comments and discussions. We also gratefully acknowledge valuable suggestions by an anonymous referee. We are indebted to Matthias Schott for his help with the numerical analysis. ∗ Correspondence to: University of Cologne, Albertus-Magnus-Platz, 50923 Cologne, Germany. Tel.: +49 0 221 470 5822; fax: +49 0 221 470 5024. E-mail address: [email protected] (A. Wambach). 1 For an alternative approach to modeling fair-minded behavior, see Bolton and
Ockenfels (2000). Linhart (2001) assumes both seller and buyer to either minimize maximum regret or maximize maximum profit. 0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.09.027
auction, a seller and a buyer independently submit a bid. Trade only occurs if the price submitted by the seller is smaller than the buyer’s bid.2 In the case without inequity aversion, inefficiency arises because in order to get a better deal, the seller tends to overstate her costs and the buyer has an incentive to understate his valuation. With sufficiently strong compassion, such a behavior would result in disutility as any additional gain is necessarily accompanied by a loss to the other party. Having a closer look at the efficiency of the double auction if compassion plays a less crucial role, we show that in the limit of strong envy, trade breaks down. If the envy factor is finite, we analyze the equilibrium in the double auction numerically. We find that an increase in the envy factor leads to bidding strategies which are further away from truth-telling. The opposite is true for an increase in the compassion factor. As a result, the lower (higher) the degree of envy (compassion), the more efficient the bargaining outcome. There exists some theoretical work on bargaining where inequity aversion is considered. These articles differ from the present work with respect to both the bargaining setup and the informational assumptions (Fehr and Schmidt, 1999; Lopomo and Ok, 2001; Ellingsen and Johannesson, 2004; Ewerhart, 2006; Montero, 2007; von Siemens, 2009).
2 Myerson and Satterthwaite (1983) show for uniformly distributed costs and valuations that the double auction is the most efficient mechanism which, however, is ex post inefficient.
A. Rasch et al. / Economics Letters 114 (2012) 178–181
The present article contributes to the literature that addresses the question to what extent inequity aversion mitigates or aggravates problems resulting from informational asymmetries. In some cases, efficiency decreases due to inequity aversion (see, eg., Grund and Sliwka, 2005; Bartling and von Siemens, 2010b; Englmaier and Wambach, 2010) whereas informational problems may also become less severe if inequity aversion is prevalent (Kragl and Schmid, 2009; Bartling and von Siemens, 2010a; Küçükşenel, 2010). Our paper is in particular related to the small part of the literature which explicitly distinguishes between the potentially different effects of envy and compassion (Grund and Sliwka, 2005; Montero, 2007; Rey-Biel, 2008). 2. Model and results We analyze a bargaining problem where a seller produces a good a buyer wants to buy. Denote by c the costs the seller has to incur when producing the good and by v the value of the object for the buyer. Both costs and valuations are assumed as independent random variables which are distributed over a given interval c ∈ [c , c¯ ] and v ∈ [v, v¯ ] (with c < v¯ ). f (v) and g (c ) are the respective probability density functions which are assumed continuous and positive on their domains. The corresponding distribution functions are F (c ) and G(v). Costs and valuations are private information and hence only known to the respective party. Consider a mechanism ζ = (x(ˆc , vˆ ), p(ˆc , vˆ )) where x(ˆc , vˆ ) ∈ [0, 1] is the allocation rule and p(ˆc , vˆ ) is the payment rule conditional on trade taking place. Both rules depend on the messages cˆ and vˆ sent by the seller and buyer, respectively. Suppose the other party tells the truth, then following Fehr and Schmidt (1999), the expected utility from bargaining is given by E[US (c , cˆ )] =
v¯
∫ v
x(ˆc , v)(p(ˆc , v) − c
− α max{0, −(p(ˆc , v) − c ) + (v − p(ˆc , v))} − β max{0, (p(ˆc , v) − c ) − (v − p(ˆc , v))})g (v)dv and E[UB (v, vˆ )] =
c¯
∫
x(c , vˆ )(v − p(c , vˆ ) c
− α max{0, −(v − p(c , vˆ )) + (p(c , vˆ ) − c )} − β max{0, (v − p(c , vˆ )) − (p(c , vˆ ) − c )})g (c )dc where subscripts S and B denote the seller and buyer, respectively. If trade takes place, the first two terms in the brackets represent the gross gains of trade to the seller and the buyer. The third and the fourth terms reflect the disutility the parties incur from an unfair division of the surplus where the third term gives the disadvantage weighted with the envy factor α . The fourth term accounts for the advantage to the seller or buyer weighted with the compassion factor β . It is assumed that α ≥ β and 0 ≤ β ≤ 1.3 Due to the revelation principle, we restrict our attention to incentive-compatible direct mechanisms (Myerson, 1981). The maximization problem can be written as a weighted sum of the individual expected utilities
179
as well as the individual-rationality constraints E[US (c , c )] ≥ 0, E[UB (v, v)] ≥ 0. In what follows, we will look for a fully efficient mechanism that can be implemented when agents are inequity averse. Note that (ex post) efficiency refers to two aspects: first, the allocation is efficient whenever all possible gains from trade are realized, i.e., trade occurs for c ≤ v . Second, the payment rule must be such that agents do not incur disutility from inequity aversion, i.e., the gains from trade are shared equally. Hence, the optimal allocation and payment rules are given by x∗ (c , v) =
1 0
if c ≤ v else
(1)
.
(2)
and p∗ (c , v) =
c+v 2
As the following proposition shows, full efficiency can be implemented for general distributions of costs and valuations if and only if compassion is strong enough. Proposition 1. The mechanism ζ ∗ = (x∗ (ˆc , vˆ ), p∗ (ˆc , vˆ )) is incentive compatible and thus implements full efficiency if and only if β ≥ 1/2. Proof. See the Appendix.
The intuition behind this result is as follows. Consider the seller side: if a seller expecting the buyer to send a message equal to his valuation reveals her true costs, the optimal payment lies halfway between costs and valuation if trade occurs. Under these circumstances, both parties are equally well off for any value of both costs and valuation and there is no disutility from inequity on either side. If, on the other hand, the seller’s message is above her actual costs, this leads to a better payment from her point of view. At the same time, however, this results in an unequal final bargaining outcome. Now if compassion is of sufficiently great importance, the seller prefers to tell the truth concerning her costs. Next consider the double auction as a possible mechanism. In the double auction, both the seller and the buyer submit bids. Trade occurs whenever the seller’s bid bS is equal to or smaller than the buyer’s bid bB at a price that is equal to the average bid, i.e., p = (bS +bB )/2. Hence, if the seller and the buyer submit bids equal to their true costs and valuation, respectively, the allocation and the payment are identical to expressions (1) and (2) of the optimal mechanism. Therefore, the next result follows immediately from Proposition 1. Corollary 1. The double auction implements full efficiency if β ≥ 1/2.4 , 5 We use the double auction to address the case where compassion is weak. As the following proposition points out, the situation becomes very much different if envy is strong. Proposition 2. Suppose that v ≤ c and/or v¯ ≤ c¯ and that β < 1/2. Then, in the limit where α → ∞, trade occurs with probability zero in equilibrium in the double auction.
max λE[US (c , c )] + (1 − λ)E[UB (v, v)] x ,p
subject to the incentive-compatibility constraints E[US (c , c )] ≥ E[US (c , cˆ )], E[UB (v, v)] ≥ E[UB (v, vˆ )]
3 For a discussion of these assumptions, see Fehr and Schmidt (1999).
4 In their setup, Chatterjee and Samuelson (1983) show that truth-telling is also an equilibrium if both sides are very (infinitely) risk averse. 5 Note that truth-telling with its fair outcome is not the only strategy which leads to a fair allocation with trade in the double auction. Such an allocation can also be obtained if both sides bid according to a strategy with slope 1, i.e., bS (c ) = c + ∆ and bB (v) = v − ∆ with ∆ ≶ 0. See Rasch et al. (2011) for details.
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A. Rasch et al. / Economics Letters 114 (2012) 178–181
Fig. 1. Equilibrium bidding strategies for the seller for different values of α and β : (a) α = β = 0 (dashed line), (b) β ≥ 1/2 (dashed line), (c) α = 1/2, β = 0.495, (d) α = 1/4, β = 0, and (e) α = 10, β = 0.
Fig. 2. Trade regions (upper left corner) for different values of α and β : (a) α = β = 0 (dashed line), (b) β ≥ 1/2 (dashed line), (c) α = 1/2, β = 0.495, (d) α = 1/4, β = 0, and (e) α = 10, β = 0.
Proof. See the Appendix.
omitted. Consider the seller who deviates from the truth-telling strategy and sends a message cˆ > c. Assume that the buyer sticks to the truth-telling strategy, i.e., vˆ = v . Hence, given expressions (1) and (2), the seller’s expected utility is given by
If compassion is weak, trade no longer takes place as envy becomes extremely important. This is rather intuitive. If an equal split is not possible, then one trading partner will be worse off than the other. Now as the importance of envy increases, the less attractive will be the trade outcome for the side that is worse off. The only ‘fair’ outcome which is then reachable is the no-trade outcome.6 Next consider the case where β < 1/2 and α is finite. As it turns out, a general solution to the problem is not analytically tractable. Therefore, we resort to numerical simulations. Figs. 1 and 2 illustrate the seller’s equilibrium bidding strategies in the double auction7 for different parameter values of inequity aversion as well as the resulting trade region (for uniformly distributed costs and valuations). The buyer’s strategies are analogous. If the costs for supplying the product are high, the seller bids more aggressively (cases (d) and (e)) than without inequity aversion (case (a)) due to envy. Now if envy becomes more important, the bid will be even higher (compare case (d) to case (e)). On the other hand, if the costs are low, obtaining more trade through a lower bid is optimal compared to the situation without inequity aversion (cases (c) and (d)). As the importance of compassion rises (case (c)), the seller bids less aggressively in order to avoid being better off. The numerical analysis shows that an increase in the degree of envy leads to less trade (compare cases (d) and (e)). This is in line with Proposition 2 where, in the limit, trade breaks down completely. On the other hand, an increase in the degree of compassion results in more trade (compare cases (c) and (d)). Moreover, as shown in Corollary 1, if β converges to 1/2, truthtelling becomes optimal and trade occurs whenever it is efficient (see case (b)). Appendix A.1. Proof of Proposition 1
E[US (c , cˆ )] =
v¯
∫
cˆ + v 2
cˆ
∫ v¯
−β
=
cˆ + v 2
cˆ
=
1
∫
− c g (v)dv cˆ + v 2
cˆ
v¯
∫
cˆ + v −c − v− g (v)dv
− c − β cˆ − c
2
g (v)dv
v¯
v g (v)dv 1 + cˆ − β − c (1 − β) (1 − G(ˆc )). 2 cˆ
2
Note that if trade occurs, this will lead to a price greater than (c + v)/2, i.e., only the advantage term is relevant for the seller. Trade occurs whenever v ≥ cˆ . It follows that
∂ E[US (c , cˆ )] = ∂ cˆ
1 2
− β (1 − G(ˆc )) − (ˆc − c )(1 − β)g (ˆc ).
For truth-telling to be optimal, ∂ E[US (c , cˆ )]/∂ cˆ < 0 ∀ˆc > c must hold. Since 1 − G(ˆc ) ≥ 0, this is always true whenever β ≥ 1/2. On the other hand, if β < 1/2, then for cˆ = c, it follows that
∂ E[US (c , cˆ )] ∂ cˆ
> 0, cˆ =c
i.e., the mechanism is not incentive compatible. Now consider the case where cˆ < c. Under the rule that gains from trade are split equally, such a message not only leads to a reduction in the seller’s payoff by passing more to the buyer but also yields additional disutility by favoring the buyer. Therefore, a message cˆ < c is never optimal.
Proof. We prove the proposition for the seller only. The case for the buyer can be analyzed in an analogous fashion and is therefore
A.2. Proof of Proposition 2
6 If v > c, equilibria where b (c ) = c + ∆ and b (v) = v − ∆ with ∆ > 0 exist S B in the double auction even for β < 1/2 (Rasch et al., 2011). 7 Note that Fig. 1 only shows the equilibrium bidding behavior for those seller
Proof. We prove the proposition by contradiction. Assume that there exists an equilibrium where trade occurs with strictly positive probability for any value of α . From Proposition 1 above and Proposition 2 in Rasch et al. (2011), we know that fair allocations are not an equilibrium under the assumptions of Proposition 2.
types who trade. Seller types who do not trade may submit any bid as long as bS (c ) > bB (v) ∀v holds (e.g., bS (c ) = 1).
A. Rasch et al. / Economics Letters 114 (2012) 178–181
Denote the sets of types who trade by HS and HB . A seller with costs c ∈ HS who trades with buyers of types v ∈ HB may be worse off, equally well off, or better off compared to her trading partner. We thus define the following sets which correspond to these cases: AS (c , α) := {v ∈ HB |(c + v)/2 > p(c , v, α)}, BS (c , α) := {v ∈ HB |(c + v)/2 = p(c , v, α)}, and CS (c , α) := {v ∈ HB |(c + v)/2 < p(c , v, α)} where p(c , v, α) is the bargaining price that depends on α . The seller then gets an expected utility of E[US (c , v)] = E[US (c , v)|v ∈ AS (c , α)]
+ E[US (c , v)|v ∈ BS (c , α)] + E[US (c , v)|v ∈ CS (c , α)]. Consider first the case where limα→∞ Pr(AS (c , α)) = 0 ∀c ∈ HS and limα→∞ Pr(AB (v, α)) = 0 ∀v ∈ HB which implies that types who trade must bid their costs or valuations (±∆ with ∆ > 0), respectively. This, however, cannot be an equilibrium as shown in Proposition 2 in Rasch et al. (2011). Assume therefore that limα→∞ Pr(AS (c , α)) ≥ kS with kS > 0. The expected utility for a seller who trades amounts to E[US (c , v)] =
∫ AS (c ,α)
−α
(p(c , v, α))g (v)dv
∫ AS (c ,α)
(c + v − 2p(c , v, α))g (v)dv
+ E[US (c , v)|v ∈ BS (c , α)] + E[US (c , v)|v ∈ CS (c , α)]. Given that Pr(AS (c , α)) ≥ kS , it then holds that α → ∞ ⇒ E[US (c , v)] < 0 if and only if A (c ,α) (c + v − 2p(c , v, α))g (v)dv S > 0 for α → ∞. Assume that the opposite holds, i.e.,
∫ AS (c ,α)
(c + v − 2p(c , v, α))g (v)dv = 0 for α → ∞.
Let A′S (c , α) := {v ∈ AS (c , α)|c + v − 2p(c , v, α) > ϵ} ⊂ AS (c , α). We can thus conclude that
∫ lim
α→∞ A′ (c ,α) S
(c + v − 2p(c , v, α))g (v)dv = 0 ∀ϵ > 0
and hence lim Pr(A′S (c , α)) = 0 ∀ϵ > 0.
α→∞
Then, assuming that ‖ · ‖∞ is the supremum norm for set AS (c , α), it must hold that lim ‖c + v − 2p(c , v, α)‖∞ = 0
α→∞
a.s.
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Hence, it is true that ∀ϵ > 0 ∃α˜ such that ∀α > α˜ ,
‖c + v − 2p(c , v, α)‖∞ < ϵ a.s. which is equal to sup
v∈AS (c ,α)
|c + v − 2p(c , v, α)| < ϵ a.s.
It thus follows that ∀ϵ > 0 ∃α˜ such that ∀α > α˜ and for almost all v ∈ AS (c , α), it must hold that
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