Parties in a bargaining situation may perceive guilt, a utility loss caused by receiving the larger share that is modeled in some social preferences. I extend Rubinstein (1982)'s solution of the open-ended alternating-offer bargaining problem for self-interested bargainers to a game with equally patient bargainers that exhibit a similar degree of guilt. The bargaining parties still reach agreement in the first period. If guilt is strong, they split the bargaining surplus equally. In contrast, if guilt is weak, the bargaining outcome is tilted away from the Rubinstein division towards a more unequal split. As both bargainers sensation of guilt diminishes, the bargaining outcome converges to the Rubinstein division

Kohler, Stefan

Munich RePEc Personal Archive

MPRAMunich Personal RePEc ArchiveGuilt causes equal or unequal division inalternating-offer bargainingStefan KohlerInstitute for Social Medicine, Epidemiology and Health Economics,Charite´ University Medical Center, Free University and HumboldtUniversity of Berlin, Germany10. September 2012Online at http://mpra.ub.uni-muenchen.de/40760/MPRA Paper No. 40760, posted 12. September 2012 12:48 UTCGuilt causes equal or unequal division inalternating-offer bargainingStefan Kohler∗AbstractParties in a bargaining situation may perceive guilt, a utility loss caused by receiving thelarger share that is modeled in some social preferences. I extend Rubinstein (1982)’s solutionof the open-ended alternating-offer bargaining problem for self-interested bargainers to agame with equally patient bargainers that exhibit a similar degree of guilt. The bargainingparties still reach agreement in the first period. If guilt is strong, they split the bargainingsurplus equally. In contrast, if guilt is weak, the bargaining outcome is tilted away fromthe Rubinstein division towards a more unequal split. As both bargainers sensation of guiltdiminishes, the bargaining outcome converges to the Rubinstein division.Keywords: alternating offers, bargaining, bargaining power, behavioral economics, equity, fair-ness, guilt, inequality aversion, negotiation, social preferencesJEL classifications: C72, C78, D03, D31, D63, D641 IntroductionIn bargaining encounters, a mutually beneficial outcome can be realized if the parties partici-pating in the bargaining process reach agreement. Rubinstein (1982) proposed a seminal frame-work to investigate open-ended alternating-offer bargaining situations. The framework re-vealed the interdependence of parties’ intertemporal strategic considerations, and it suggestedan explicit solution for purely self-interested parties which depicts a focal division from the setof all possible pareto-efficient agreements. Supplementary, this paper studies the additionalimpact of a psychological element in the utility, suggested, for instance, by Von Neumann &Morgenstern (1944), on the bargaining process and outcome. I extend Rubinstein’s solution ofthe alternating-offer bargaining problem to similar, guilt-perceiving parties that, to some de-gree, dislike receiving a larger share. Guilt is modeled as an asymmetric form of inequalityaversion and affects the bargainers’ attitudes towards disagreement.The relative strength of guilt in comparison to bargaining parties’ self-interest impacts thebargaining outcome in two ways: High guilt triggers an equal division because the bargainers’utility decreases when receiving more than half. Low guilt diminishes the marginal utility ofincome, but preserves its positive marginal utility. Own low guilt, ceteris paribus, weakensthe own bargaining position. Yet, the same degree of low guilt in both bargainers, overall,∗September 10, 2012. Institute for Social Medicine, Epidemiology and Health Economics, Charité UniversityMedical Center, Free University and Humboldt University of Berlin, Germany; Email: stefan.kohler@charite.de,Homepage: http://epidemiologie.charite.de. I have benefited from helpful and constructive comments by KarlSchlag, Pascal Courty, Simon Gächter, Jordi Brandts and anonymous reviewers. I gratefully acknowledge the fi-nancial support provided by the European University Institute and the German Academic Exchange Service at anearly stage of this research.1helps the proposing bargainer to take a larger share than predicted by Rubinstein. The para-dox as to why feeling guilty about a larger share can result in a more unequal division than thebargaining of self-interested parties is driven by the weakened bargaining position of the dis-advantaged bargainer. The disadvantaged bargainer compares accepting a share smaller thanhalf to proposing a share larger than half in the subsequent period, the utility of which is di-minished by guilt. As low guilt maintains a positive marginal utility of income, the proposingbargainer exploits the lowered value of the accepting bargainer’s outside option by increasinghis demand.In alternating-offer bargaining, low guilt has the opposite effects of envy which are studiedby Kohler (2012). Envy reinforces the bargaining position of a bargainer. If the two bargain-ers are similarly envious, then the bargaining outcome departs from the Rubinstein divisionconverging toward an equal split. Guilt shares some features with altruism because a guilt per-ceiving person feels altruistic towards others until the advantageous situation disappears (e.g.,Kohler 2011). Altruism may be beneficial when there is competition for bargaining partners.It may be detrimental if bargainers discriminate towards whom they feel altruistic (Montero2008).Empirical evidence of guilt in addition to envy in bargaining experiments is mixed. Esti-mating average inequality aversion as introduced by Bolton & Ockenfels (2000) in a structuralmodel of bounded rationality based on data from an ultimatum game field experiment, Kohler(2008) finds the data is better explained by a model of guilt and envy, i.e., symmetric inequal-ity aversion, than by a model of envy, i.e., asymmetric inequality aversion. Also De Bruyn &Bolton (2008) discuss that symmetric inequality aversion may statistically improve the fit ofdifferent experimental bargaining data in a meta-analyses, but they prefer a model of envy topredict bargaining behavior because, inter alia, it provides better out-of-sample forecasts. Bet-ter out of-sample forecasts were further reported for the Bolton & Ockenfels than for the Fehr &Schmidt (1999) specification of inequality aversion. This theoretical paper, employs a restrictedFehr & Schmidt utility function, which is a linear variant of the Bolton & Ockenfels model ofinequality aversion in two-player games.Section 2 introduces the alternating-offer bargaining problem with guilt-perceiving parties.In section 3, I derive the bargaining outcome. Section 4 concludes.2 Bargaining modelTwo bargainers i, j ∈ {b, s}, called seller and buyer, have to reach an agreement on the partitionof a surplus of size one which depreciates after any disagreement. Bargaining takes place atperiods of time t = 1, 2, ..., T. Depreciation is modeled by assigning a common discount factorδ to the two bargainers. By naming a partition pt ∈ (0, 1] in odd periods, the seller demandsshare pt and offers share (1− pt) that the buyer can accept or reject. In even periods, the buyerproposes a partition pt to the seller that he can accept or reject. If a partition is accepted thegame ends in period T. This bargaining outcome is denoted (pT, T).Assuming complete information in this bargaining problem, Rubinstein (1982) has shownthe existence of a unique subgame perfect equilibrium (SPE) under generic preference as-2sumptions.1 For preferences ui (xi) = xi, where utility is derived from own payoff xi, Rubin-stein derived an explicit solution, in which the seller proposes and the buyer accepts partitionp∗ = 11+δ ∈ (0, 1] in period 1. The equilibrium outcome is supported by the bargainers’ similarstrategies: Bargainer i always demands the equilibrium share p∗, when it his turn to make aproposal, otherwise accepts any share equal or greater than δp∗ and refuses any smaller share.The demand of p∗ is the highest share that is accepted by the other bargainer j. Bargainer i can-not gain by asking a lower share, for it too will be accepted. Stipulating a higher (and rejected)share and waiting to accept bargainer j’s counteroffer in the next period hurts bargainer i asδ (1− p∗) = δ2 p∗ < p∗.This paper builds on the generality of Rubinstein’s framework and investigates the strategicbehavior of bargainers who care, to some extend, about relative as well as absolute payoffin the described bargaining process. Relative payoff hereby means bargainers compare theirown benefit xi from accepting a certain partition to the benefit of the other bargainer xj, andput weight put a common weight β = βs ≡ βb ∈ [0, 1) on the difference whenever the ownbenefit is higher. This relative concern is interpreted as guilt. Explicitly, I assume that the utilityfunction of the bargainers is given by:ui(xi, xj)= xi − βi max{xi − xj, 0}These preferences are an asymmetric version of inequality aversion as originally put forwardin Fehr & Schmidt (1999) and extended by altruism in Kohler (2011). Inequality aversion con-sistently predicts a rich set of stylized experimental behavior (e.g., Cooper & Kagel n.d.; Fehr &Schmidt 1999). The asymmetric guilt preferences fulfill Rubinstein’s requirements for the exis-tence of a unique bargaining outcome. Throughout, us(pt) := us(pt, 1− pt) denotes the seller’sutility and ub(pt) := ub(1− pt, pt) the corresponding buyer’s utility if a proposed partition ptis accepted in period t.3 Subgame perfect equilibriumProposition 1. The alternating-offer bargaining problem with guilt-perceiving discounting bargainershas a unique SPE. The seller immediately proposes the partition p∗ = 1−βδ1+δ(1−2β) if guilt is low, i.e.,β < 0.5, or the partition p∗ = 0.5 that divides the surplus equally if guilt is high, i.e., β ≥ 0.5. Therespective partition is accepted immediately.The proof of proposition 1 is divided in two parts. For low guilt β < 0.5, the first part of theproof is based on Shaked & Sutton (1984) who applied backwards induction in a truncation ofthe infinite horizon game: The beginning of the infinite horizon game is equal to its subgamein the third round, should it be reached. In odd periods, the seller is proposing and thenbargainers alternate in making subsequent offers until an agreement is reached. For high guilt1(i) Trade is desirable: ∂u(x)∂x > 0, (ii) Time is valuable: δ < 1, (iii) Continuity: limx→x u (x) = limx←x u (x), (iv)Stationarity: preferences are time independent, (v) The larger the share the more compensation a bargainer needsfor a delay of one period to be immaterial to him. Strategies are said to constitute a SPE if, in every subgame, thestrategies relating to that subgame form a Nash equilibrium. In a SPE, a bargainer will agree to a proposal if it offersat least as much as he will obtain in the future given the strategies of both bargainers. Rubinstein (1982) states theprecise definition.3β ≥ 0.5, the argument of second part of the proof is based on the negative marginal utility ofown income.Proof. For β < 0.5, I assume that the proposing bargainer can always claim the larger shareon the equilibrium path, i.e., p1 ≥ 0.5, p2 ≤ 0.5, p3 ≥ 0.5. This assumption is shown tobe valid after deriving the equilibrium outcome. The period 3 subgame begins with a suc-cessful proposal p3 ∈ [0.5, 1] by a guilt-perceiving seller. Consequently, the lowest sharep2 = δ (p3 + βs (1− 2p3)) that is accepted by the seller in period 2 gives him the equiva-lent of his outside option, the discounted period 3 utility. Similarly, the highest share p1 =1− δ ((1− p2)− βb (1− 2p2)) that is accepted by the buyer in period 1 gives him the equiva-lent of his outside option, the discounted period 2 utility. Indifferent bargainers are assumed toaccept the proposed partition. As ub(p2) ≥ δub(p3) and us(p1) ≥ δus(p2), the buyer and sellerprefer proposing the agreeable partitions that maximize their utility to disagreement with thesubsequent counteroffer.Since the game in period 3 is identical to the game in period 1, the unique fixed pointp∗ := p1 (p3) ≡ p3 defines the equilibrium partition:p∗ =1− δ ((1− δβs)− βb (1− 2δβs))1− δ (δ (1− 2βs)− 2δβb (1− 2βs)) =1− βδ1+ δ (1− 2β)As only p2 and p1 maximize the utility of the bargainer proposing the partition, there is noother SPE.The partial derivatives of the equilibrium partition p∗ with respect to guilt ∂p∗/∂β ≥ 0and the discount factor ∂p∗/∂δ < 0 are weakly positive and negative, respectively. Hence,the infimum of p∗ is lim(β,δ)→(0,1) p∗ which evaluates to 0.5. Similarly, as ∂p2/∂β ≤ 0 and∂p2/∂δ > 0, the supremum of p2 (p∗) is lim(β,δ)→(0,1) p2 which also evaluates to 0.5. Therefore,the assumed advantage of the proposing bargainer to receive half or more of the surplus on theequilibrium path is true. The partial derivatives are derived in appendix.If β ≥ 0.5, then ∂us/∂pt ≤ 0 for pt > 0.5. Thus, the equal division is weakly preferred toany advantageous share by the seller in any period t. Further, as ∂ub/∂pt ≤ 0 for pt ≥ 0.5, theequal division is preferred by the buyer receiving the disadvantageous share in period t. By thesame arguments the buyer also weakly prefers the equal division to any advantageous shareand the seller prefers redistribution of the surplus until its equal division is agreed. Indifferentbargainers for which β = 0.5 are assumed to choose the equal division. Thus, p∗ = 0.5 is theonly equilibrium partition. As bargainers discount it is immediately asked and agreed.4 ConclusionBargainers may incur guilt, a loss of utility if receiving an advantageous share of a surplus tobe divided, in a bargaining process. In open-ended alternating offer bargaining between twoparties with similar time and inequality preferences, strongly guilt-perceiving bargainers gainutility from reducing an advantageous situation until the inequality between the bargainers iseliminated. Therefore, in the presence of sufficient guilt, the unique bargaining outcome is theimmediate acceptance of an equal division.4In contrast, bilateral low guilt materially benefits the proposing bargainer. If the bargainingparties perceive guilt only to such an extend that their utility remains increasing in the ownpayoff despite increasing inequality, then the impact of guilt results in a more unequal divisionthan predicted by Rubinstein (1982) for purely self-interest bargainers. In spite of the firstmovers material gain in the equilibrium, low guilt diminishes the utility of both bargainersin comparison to the utility that purely self-interested parties derive from their more equalbargaining outcome (see appendix).For a low strength of guilt, the partial derivatives of the equilibrium partition imply thatthe first mover’s share is increasing in the common strength of guilt. On its own, the firstand the second mover’s guilt have opposite effects on the equilibrium division. The share ofeach bargainer decreases in the strength of the own guilt. Like in Rubinstein (1982)’s solutionwithout guilt, agreement is immediate and the higher a common discount factor, the moreequal will be the agreed division. Taking the limits of the equilibrium partition for low guiltshows that the equilibrium partition is between the equal division and one.Notwithstanding the opposite directions of the effect of guilt on the alternating-offer bar-gaining outcome, the bargaining of symmetrically guilt-perceiving bargainers is never equiva-lent to the bargaining of impatient purely self-interested parties.5 Appendix5.1 Partial derivatives and limits for low guiltIf β < 0.5, the equilibrium partition p∗ and its partial derivatives are given by:p∗ =1− δ ((1− δβs)− βb (1− 2δβs))1− δ (δ (1− 2βs)− 2δβb (1− 2βs)) =1− βδ1+ δ (1− 2β) =:ND∂p∗∂δ= − (1− β) · D (β, δ)−2 < 0∂p∗∂β= δ (1− δ) · D (β, δ)−2 ≥ 0∂p∗∂βs= −δ2 (1− δ) (1− 2βb) (1− δ+ 2δβb) · D (βs, βb, δ)−2 ≤ 0∂p∗∂βb= δ (1− δ) (1− δ+ 2δβs) · D (βs, βb, δ)−2 ≥ 0The respective signs follow from evaluating the derivatives. The limits of the equilibriumpartition p∗ are given by limδ→0 p∗ = 1, limδ→1 p∗ = 0.5 and limβ→0 p∗ = 11+δ ∈ (0.5, 1],limβ→0.5 p∗ = 1− 0.5δ ∈ (0.5, 1]. The limit values follow from evaluating the limits.If β < 0.5, the partition p2 and its partial derivatives are given by:5p2 (p∗) = δ1− β1+ δ (1− 2β) =:ND∂p2∂δ= (1− β) · D−2 > 0∂p2∂β= −δ (1− δ) · D−2 ≤ 0The respective signs follow from evaluating the derivatives. The limits of the partition p2are given by limδ→0 p2 = 0, limδ→1 p2 = 0.5 and limβ→0 p2 = δ1+δ ∈ [0, 0.5), limβ→∞ p2 = 0.5.The limit values follow from evaluating the limits.5.2 Utility in the subgame perfect equilibrium with guiltIrrespective of the strength of guilt, the utility of a guilt-perceiving seller, who perceives guiltin the equilibrium, is not higher than the utility of a purely self-interested seller that receives11+δ :us (p∗)us( 11+δ)∣∣β=0={(1−β)(1+δ)1+δ(1−2β) ∈ (0.5, 1] if β < 0.51+δ2 ∈ [0.5, 1) if β ≥ 0.5The utility of a guilt-perceiving buyer, whose utility in equilibrium is unaffected by guilt, is notlower than the utility of a purely self-interested buyer if guilt is low and higher if guilt is high:ub (p∗)ub( 11+δ)∣∣β=0={(1−β)(1+δ)1+δ(1−2β) ∈ (0.5, 1] if β < 0.51+δ2δ ∈ (1,∞] if β ≥ 0.5ReferencesBolton, G. E., & Ockenfels, A. (2000). ERC: A theory of equity, reciprocity and competition.American Economic Review, 90, 166–193.Cooper, D. J., & Kagel, J. H. (n.d.). Other regarding preferences: A selective survey of experi-mental results. In J. H. Kagel, & A. E. Roth (Eds.), Handbook of Experimental Economics, Volume2. Princeton: Princeton University Press.De Bruyn, A., & Bolton, G. E. (2008). Estimating the influence of fairness on bargaining behav-ior. Management Science, 54, 1774–1791.Fehr, E., & Schmidt, K. M. (1999). A theory of fairness, competition, and cooperation. QuarterlyJournal of Economics, 114, 817–868.Kohler, S. (2008). Inequality aversion and stochastic decision-making: Experimental evidencefrom Zimbabwean villages after land reform. In A. Innocenti, & P. Sbriglia (Eds.), Games,Rationality and Behavior: Essays on Behavioral Game Theory and Experiments (pp. 63–100). Bas-ingstoke: Palgrave McMillan.Kohler, S. (2011). Altruism and fairness in experimental decisions. Journal of Economic Behavior& Organization, 80, 101–109.6Kohler, S. (2012). Envy can promote more equal division in alternating-offer bargaining. MPRAPaper 40761, University Library of Munich, Germany.Montero, M. (2008). Altruism, spite and competition in bargaining games. Theory and Decision,65, 125–151.Rubinstein, A. (1982). Perfect equilibrium in a bargaining model. Econometrica, 50, 97–109.Shaked, A., & Sutton, J. (1984). Involuntary unemployment as a perfect equilibrium in a bar-gaining model. Econometrica, 52, 1351–1364.Von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton,New Jersey, USA: Princeton University Press.7

English

Guilt causes equal or unequal division in alternating-offer bargaining

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
Guilt causes equal or unequal division in
alternating-offer bargaining
Stefan Kohler
Institute for Social Medicine, Epidemiology and Health Economics,
Charite´ University Medical Center, Free University and Humboldt
University of Berlin
10 September 2012
Online at https://mpra.ub.uni-muenchen.de/40760/
MPRA Paper No. 40760, posted 12 September 2012 12:48 UTC
Guilt causes equal or unequal division in
alternating-offer bargaining
Stefan Kohler∗
Abstract
Parties in a bargaining situation may perceive guilt, a utility loss caused by receiving the
larger share that is modeled in some social preferences. I extend Rubinstein (1982)’s solution
of the open-ended alternating-offer bargaining problem for self-interested bargainers to a
game with equally patient bargainers that exhibit a similar degree of guilt. The bargaining
parties still reach agreement in the first period. If guilt is strong, they split the bargaining
surplus equally. In contrast, if guilt is weak, the bargaining outcome is tilted away from
the Rubinstein division towards a more unequal split. As both bargainers sensation of guilt
diminishes, the bargaining outcome converges to the Rubinstein division.
Keywords: alternating offers, bargaining, bargaining power, behavioral economics, equity, fair-
ness, guilt, inequality aversion, negotiation, social preferences
JEL classifications: C72, C78, D03, D31, D63, D64
1 Introduction
In bargaining encounters, a mutually beneficial outcome can be realized if the parties partici-
pating in the bargaining process reach agreement. Rubinstein (1982) proposed a seminal frame-
work to investigate open-ended alternating-offer bargaining situations. The framework re-
vealed the interdependence of parties’ intertemporal strategic considerations, and it suggested
an explicit solution for purely self-interested parties which depicts a focal division from the set
of all possible pareto-efficient agreements. Supplementary, this paper studies the additional
impact of a psychological element in the utility, suggested, for instance, by Von Neumann &
Morgenstern (1944), on the bargaining process and outcome. I extend Rubinstein’s solution of
the alternating-offer bargaining problem to similar, guilt-perceiving parties that, to some de-
gree, dislike receiving a larger share. Guilt is modeled as an asymmetric form of inequality
aversion and affects the bargainers’ attitudes towards disagreement.
The relative strength of guilt in comparison to bargaining parties’ self-interest impacts the
bargaining outcome in two ways: High guilt triggers an equal division because the bargainers’
utility decreases when receiving more than half. Low guilt diminishes the marginal utility of
income, but preserves its positive marginal utility. Own low guilt, ceteris paribus, weakens
the own bargaining position. Yet, the same degree of low guilt in both bargainers, overall,
∗September 10, 2012. Institute for Social Medicine, Epidemiology and Health Economics, Charité University
Medical Center, Free University and Humboldt University of Berlin, Germany; Email: stefan.kohler@charite.de,
Homepage: http://epidemiologie.charite.de. I have benefited from helpful and constructive comments by Karl
Schlag, Pascal Courty, Simon Gächter, Jordi Brandts and anonymous reviewers. I gratefully acknowledge the fi-
nancial support provided by the European University Institute and the German Academic Exchange Service at an
early stage of this research.
1
helps the proposing bargainer to take a larger share than predicted by Rubinstein. The para-
dox as to why feeling guilty about a larger share can result in a more unequal division than the
bargaining of self-interested parties is driven by the weakened bargaining position of the dis-
advantaged bargainer. The disadvantaged bargainer compares accepting a share smaller than
half to proposing a share larger than half in the subsequent period, the utility of which is di-
minished by guilt. As low guilt maintains a positive marginal utility of income, the proposing
bargainer exploits the lowered value of the accepting bargainer’s outside option by increasing
his demand.
In alternating-offer bargaining, low guilt has the opposite effects of envy which are studied
by Kohler (2012). Envy reinforces the bargaining position of a bargainer. If the two bargain-
ers are similarly envious, then the bargaining outcome departs from the Rubinstein division
converging toward an equal split. Guilt shares some features with altruism because a guilt per-
ceiving person feels altruistic towards others until the advantageous situation disappears (e.g.,
Kohler 2011). Altruism may be beneficial when there is competition for bargaining partners.
It may be detrimental if bargainers discriminate towards whom they feel altruistic (Montero
2008).
Empirical evidence of guilt in addition to envy in bargaining experiments is mixed. Esti-
mating average inequality aversion as introduced by Bolton & Ockenfels (2000) in a structural
model of bounded rationality based on data from an ultimatum game field experiment, Kohler
(2008) finds the data is better explained by a model of guilt and envy, i.e., symmetric inequal-
ity aversion, than by a model of envy, i.e., asymmetric inequality aversion. Also De Bruyn &
Bolton (2008) discuss that symmetric inequality aversion may statistically improve the fit of
different experimental bargaining data in a meta-analyses, but they prefer a model of envy to
predict bargaining behavior because, inter alia, it provides better out-of-sample forecasts. Bet-
ter out of-sample forecasts were further reported for the Bolton & Ockenfels than for the Fehr &
Schmidt (1999) specification of inequality aversion. This theoretical paper, employs a restricted
Fehr & Schmidt utility function, which is a linear variant of the Bolton & Ockenfels model of
inequality aversion in two-player games.
Section 2 introduces the alternating-offer bargaining problem with guilt-perceiving parties.
In section 3, I derive the bargaining outcome. Section 4 concludes.
2 Bargaining model
Two bargainers i, j ∈ {b, s}, called seller and buyer, have to reach an agreement on the partition
of a surplus of size one which depreciates after any disagreement. Bargaining takes place at
periods of time t = 1, 2, ..., T. Depreciation is modeled by assigning a common discount factor
δ to the two bargainers. By naming a partition pt ∈ (0, 1] in odd periods, the seller demands
share pt and offers share (1− pt) that the buyer can accept or reject. In even periods, the buyer
proposes a partition pt to the seller that he can accept or reject. If a partition is accepted the
game ends in period T. This bargaining outcome is denoted (pT, T).
Assuming complete information in this bargaining problem, Rubinstein (1982) has shown
the existence of a unique subgame perfect equilibrium (SPE) under generic preference as-
2
sumptions.1 For preferences ui (xi) = xi, where utility is derived from own payoff xi, Rubin-
stein derived an explicit solution, in which the seller proposes and the buyer accepts partition
p∗ = 11+δ ∈ (0, 1] in period 1. The equilibrium outcome is supported by the bargainers’ similar
strategies: Bargainer i always demands the equilibrium share p∗, when it his turn to make a
proposal, otherwise accepts any share equal or greater than δp∗ and refuses any smaller share.
The demand of p∗ is the highest share that is accepted by the other bargainer j. Bargainer i can-
not gain by asking a lower share, for it too will be accepted. Stipulating a higher (and rejected)
share and waiting to accept bargainer j’s counteroffer in the next period hurts bargainer i as
δ (1− p∗) = δ2 p∗ < p∗.
This paper builds on the generality of Rubinstein’s framework and investigates the strategic
behavior of bargainers who care, to some extend, about relative as well as absolute payoff
in the described bargaining process. Relative payoff hereby means bargainers compare their
own benefit xi from accepting a certain partition to the benefit of the other bargainer xj, and
put weight put a common weight β = βs ≡ βb ∈ [0, 1) on the difference whenever the own
benefit is higher. This relative concern is interpreted as guilt. Explicitly, I assume that the utility
function of the bargainers is given by:
ui
(
xi, xj
)
= xi − βi max
{
xi − xj, 0
}
These preferences are an asymmetric version of inequality aversion as originally put forward
in Fehr & Schmidt (1999) and extended by altruism in Kohler (2011). Inequality aversion con-
sistently predicts a rich set of stylized experimental behavior (e.g., Cooper & Kagel n.d.; Fehr &
Schmidt 1999). The asymmetric guilt preferences fulfill Rubinstein’s requirements for the exis-
tence of a unique bargaining outcome. Throughout, us(pt) := us(pt, 1− pt) denotes the seller’s
utility and ub(pt) := ub(1− pt, pt) the corresponding buyer’s utility if a proposed partition pt
is accepted in period t.
3 Subgame perfect equilibrium
Proposition 1. The alternating-offer bargaining problem with guilt-perceiving discounting bargainers
has a unique SPE. The seller immediately proposes the partition p∗ = 1−βδ1+δ(1−2β) if guilt is low, i.e.,
β < 0.5, or the partition p∗ = 0.5 that divides the surplus equally if guilt is high, i.e., β ≥ 0.5. The
respective partition is accepted immediately.
The proof of proposition 1 is divided in two parts. For low guilt β < 0.5, the first part of the
proof is based on Shaked & Sutton (1984) who applied backwards induction in a truncation of
the infinite horizon game: The beginning of the infinite horizon game is equal to its subgame
in the third round, should it be reached. In odd periods, the seller is proposing and then
bargainers alternate in making subsequent offers until an agreement is reached. For high guilt
1(i) Trade is desirable: ∂u(x)∂x > 0, (ii) Time is valuable: δ < 1, (iii) Continuity: limx→x u (x) = limx←x u (x), (iv)
Stationarity: preferences are time independent, (v) The larger the share the more compensation a bargainer needs
for a delay of one period to be immaterial to him. Strategies are said to constitute a SPE if, in every subgame, the
strategies relating to that subgame form a Nash equilibrium. In a SPE, a bargainer will agree to a proposal if it offers
at least as much as he will obtain in the future given the strategies of both bargainers. Rubinstein (1982) states the
precise definition.
3
β ≥ 0.5, the argument of second part of the proof is based on the negative marginal utility of
own income.
Proof. For β < 0.5, I assume that the proposing bargainer can always claim the larger share
on the equilibrium path, i.e., p1 ≥ 0.5, p2 ≤ 0.5, p3 ≥ 0.5. This assumption is shown to
be valid after deriving the equilibrium outcome. The period 3 subgame begins with a suc-
cessful proposal p3 ∈ [0.5, 1] by a guilt-perceiving seller. Consequently, the lowest share
p2 = δ (p3 + βs (1− 2p3)) that is accepted by the seller in period 2 gives him the equiva-
lent of his outside option, the discounted period 3 utility. Similarly, the highest share p1 =
1− δ ((1− p2)− βb (1− 2p2)) that is accepted by the buyer in period 1 gives him the equiva-
lent of his outside option, the discounted period 2 utility. Indifferent bargainers are assumed to
accept the proposed partition. As ub(p2) ≥ δub(p3) and us(p1) ≥ δus(p2), the buyer and seller
prefer proposing the agreeable partitions that maximize their utility to disagreement with the
subsequent counteroffer.
Since the game in period 3 is identical to the game in period 1, the unique fixed point
p∗ := p1 (p3) ≡ p3 defines the equilibrium partition:
p∗ =
1− δ ((1− δβs)− βb (1− 2δβs))
1− δ (δ (1− 2βs)− 2δβb (1− 2βs)) =
1− βδ
1+ δ (1− 2β)
As only p2 and p1 maximize the utility of the bargainer proposing the partition, there is no
other SPE.
The partial derivatives of the equilibrium partition p∗ with respect to guilt ∂p∗/∂β ≥ 0
and the discount factor ∂p∗/∂δ < 0 are weakly positive and negative, respectively. Hence,
the infimum of p∗ is lim(β,δ)→(0,1) p∗ which evaluates to 0.5. Similarly, as ∂p2/∂β ≤ 0 and
∂p2/∂δ > 0, the supremum of p2 (p∗) is lim(β,δ)→(0,1) p2 which also evaluates to 0.5. Therefore,
the assumed advantage of the proposing bargainer to receive half or more of the surplus on the
equilibrium path is true. The partial derivatives are derived in appendix.
If β ≥ 0.5, then ∂us/∂pt ≤ 0 for pt > 0.5. Thus, the equal division is weakly preferred to
any advantageous share by the seller in any period t. Further, as ∂ub/∂pt ≤ 0 for pt ≥ 0.5, the
equal division is preferred by the buyer receiving the disadvantageous share in period t. By the
same arguments the buyer also weakly prefers the equal division to any advantageous share
and the seller prefers redistribution of the surplus until its equal division is agreed. Indifferent
bargainers for which β = 0.5 are assumed to choose the equal division. Thus, p∗ = 0.5 is the
only equilibrium partition. As bargainers discount it is immediately asked and agreed.
4 Conclusion
Bargainers may incur guilt, a loss of utility if receiving an advantageous share of a surplus to
be divided, in a bargaining process. In open-ended alternating offer bargaining between two
parties with similar time and inequality preferences, strongly guilt-perceiving bargainers gain
utility from reducing an advantageous situation until the inequality between the bargainers is
eliminated. Therefore, in the presence of sufficient guilt, the unique bargaining outcome is the
immediate acceptance of an equal division.
4
In contrast, bilateral low guilt materially benefits the proposing bargainer. If the bargaining
parties perceive guilt only to such an extend that their utility remains increasing in the own
payoff despite increasing inequality, then the impact of guilt results in a more unequal division
than predicted by Rubinstein (1982) for purely self-interest bargainers. In spite of the first
movers material gain in the equilibrium, low guilt diminishes the utility of both bargainers
in comparison to the utility that purely self-interested parties derive from their more equal
bargaining outcome (see appendix).
For a low strength of guilt, the partial derivatives of the equilibrium partition imply that
the first mover’s share is increasing in the common strength of guilt. On its own, the first
and the second mover’s guilt have opposite effects on the equilibrium division. The share of
each bargainer decreases in the strength of the own guilt. Like in Rubinstein (1982)’s solution
without guilt, agreement is immediate and the higher a common discount factor, the more
equal will be the agreed division. Taking the limits of the equilibrium partition for low guilt
shows that the equilibrium partition is between the equal division and one.
Notwithstanding the opposite directions of the effect of guilt on the alternating-offer bar-
gaining outcome, the bargaining of symmetrically guilt-perceiving bargainers is never equiva-
lent to the bargaining of impatient purely self-interested parties.
5 Appendix
5.1 Partial derivatives and limits for low guilt
If β < 0.5, the equilibrium partition p∗ and its partial derivatives are given by:
p∗ =
1− δ ((1− δβs)− βb (1− 2δβs))
1− δ (δ (1− 2βs)− 2δβb (1− 2βs)) =
1− βδ
1+ δ (1− 2β) =:
N
D
∂p∗
∂δ
= − (1− β) · D (β, δ)−2 < 0
∂p∗
∂β
= δ (1− δ) · D (β, δ)−2 ≥ 0
∂p∗
∂βs
= −δ2 (1− δ) (1− 2βb) (1− δ+ 2δβb) · D (βs, βb, δ)−2 ≤ 0
∂p∗
∂βb
= δ (1− δ) (1− δ+ 2δβs) · D (βs, βb, δ)−2 ≥ 0
The respective signs follow from evaluating the derivatives. The limits of the equilibrium
partition p∗ are given by limδ→0 p∗ = 1, limδ→1 p∗ = 0.5 and limβ→0 p∗ = 11+δ ∈ (0.5, 1],
limβ→0.5 p∗ = 1− 0.5δ ∈ (0.5, 1]. The limit values follow from evaluating the limits.
If β < 0.5, the partition p2 and its partial derivatives are given by:
5
p2 (p∗) = δ
1− β
1+ δ (1− 2β) =:
N
D
∂p2
∂δ
= (1− β) · D−2 > 0
∂p2
∂β
= −δ (1− δ) · D−2 ≤ 0
The respective signs follow from evaluating the derivatives. The limits of the partition p2
are given by limδ→0 p2 = 0, limδ→1 p2 = 0.5 and limβ→0 p2 = δ1+δ ∈ [0, 0.5), limβ→∞ p2 = 0.5.
The limit values follow from evaluating the limits.
5.2 Utility in the subgame perfect equilibrium with guilt
Irrespective of the strength of guilt, the utility of a guilt-perceiving seller, who perceives guilt
in the equilibrium, is not higher than the utility of a purely self-interested seller that receives
1
1+δ :
us (p∗)
us
( 1
1+δ
)∣∣
β=0
=
{
(1−β)(1+δ)
1+δ(1−2β) ∈ (0.5, 1] if β < 0.5
1+δ
2 ∈ [0.5, 1) if β ≥ 0.5
The utility of a guilt-perceiving buyer, whose utility in equilibrium is unaffected by guilt, is not
lower than the utility of a purely self-interested buyer if guilt is low and higher if guilt is high:
ub (p∗)
ub
( 1
1+δ
)∣∣
β=0
=
{
(1−β)(1+δ)
1+δ(1−2β) ∈ (0.5, 1] if β < 0.5
1+δ
2δ ∈ (1,∞] if β ≥ 0.5
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7


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Guilt causes equal or unequal division in alternating-offer bargaining

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