Axiomatic and GameTheoretic Analysis of Bankruptcy and Taxation Problems a Survey William Thomson This version August I am grateful to Bettina Klaus Eiichi Miyagawa Juan MorenoTernero Anne van den Nouweland James Schummer Oscar Volij and esp ecially Nir Dagan and ChunHsien Yeh for their very useful comments I also thank a referee for detailed comments and the NSF for its supp ort under grant SES and SBR Abstract When a rm go es bankrupt how should its liquidation value be divided among its creditors This essayisanintro duction to the lit erature devoted to the formal analysis of such problems We present the rules that are commonly used in practice or discussed in theoret ical work We show howmany can be obtained by applying solution concepts develop ed in co op erative game theory for bargaining games and for coalitional games We formulate prop erties of rules rst when the p opulation of agents is xed then when it may vary com pare the rules on the basis of these prop erties and search for rules satisfying the greatest number of the prop erties together We mo del the resolution of conicting claims as strategic games and extend the mo del to handle surplus sharing and situations in which the feasible set is sp ecied in utilityspace Identifying wellb ehaved taxation rules is formally identical to identifying rules to reconcile conicting claims and all of the results we present can b e reinterpreted in that context Keywords Axiomatic analysis Bankruptcy Claims problems Prop ortional rule Talmud rule Constrained equal awards rule Con strained equal losses rule JEL Classication numb ers CDD William Thomson Department of Economics University of Ro chester Ro chester New York Tel fax email wthtroiccro chesteredu Contents Intro duction Claims problems and division rules An inventory of rules Relating division rules and solution concepts of the theory of co op erative games Bargaining solutions Solutions to coalitional games Prop erties of rules the xedp opulation case Basic prop erties Monotonicity requirements Indep endence additivityand related prop erties Op erators Prop erties of rules the variablep opulation case Population monotonicity Replication invariance Consistency Average consistency Merging and splitting claims Op erators Multiple parameter changes Strategic mo dels Extensions of the basic mo del Surplussharing Nontransferable utility problems Mo del with group constraints Exp erimental testing Conclusion References Intro duction When a rm go es bankrupt what is the fair way of dividing its liquidation value among its creditors This essay is an intro duction to the literature devoted to the formal analysis of problems of this kind whichwe call claims problems The ob jective of this literature which originates in a fundamental pap er by ONeill is to identify wellb ehaved rules for asso ciating with each claims problem a division between the claimants of the amount available We rst present several rules that are commonly used in practice or dis cussed in theoretical work We then formulate a number of app ealing prop erties that one may want rules to satisfy compare the rules on the basis of these prop erties and identify rules satisfying various combinations of the prop erties Indeed the axiomatic metho d underlies most of the developments on which we rep ort here and they illustrate the increasingly imp ortant role the metho d has been playing in the design of allo cation rules The rapid progress witnessed in the literature on the adjudication of conicting claims is largely due to researchers being able to draw on the conceptual appara tus and the pro of techniques elab orated in the axiomatic analysis of other mo dels running the gamut from abstract mo dels of game theory and so cial choice to concrete mo dels of resource allo cation We do not limit ourselves to axiomatic studies however We also showhow the to ols of co op erative game theory from b oth the theory of bargaining and the theory of coalitional games can be used to dene rules and we discuss a variety of strategic approaches The b estknown rule is the prop ortional rule which cho oses awards pro p ortional to claims Prop ortionality is in fact often taken as the denition of fairness for claims problems but we will challenge this p osition and start from more elementary considerations An imp ortant source of inspiration for the research we describ e is the Talmud in which several numerical ex amples are discussed and recommendations are made for them that conict with prop ortionality Can these recommendations be rationalized by means of wellb ehaved rules Among all existing rules are there grounds for prefer ring some to the others Are there yet other rules that deserve our attention Finallywe consider extensions of the mo del in particular some covering situations where the amount to divide is more than sucient to honor all the claimsthis is the problem of surplus sharingand mo dels where the data are sp ecied in utility space and the upp er b oundary of the feasible set is not restricted to b e contained in a hyp erplane normal to a vector of ones We close this intro duction by noting that the problem of assessing taxes as a function of incomes when the total tax to b e collected is xed is formally identical to the problem of adjudicating conicting claims All of the results we present can be reinterpreted in that context and more generally in the context of the assessment of liabilities An imp ortant question that we will not address is the extent to which the choice of particular division rules aects agents incentives to make com mitments that one party may in the end be unable to honor In the context of bankruptcy these are the incentives to loan and to b orrow In many of the other applications the parameters of the problems to be solved also result from decisions that agents have made and whatever rule is used at the division stage will in general have had an eect on these earlier choices In order to handle these kinds of issues we would need to embed division rules in a more complete mo del in which risktaking eort and other vari ables chosen by agents such as lenders b orrowers tax payers government agencies and others are explicitly describ ed sto chastic returns to economic activities are factored in and so on But the theory develop ed here which ig nores incentives is a necessary comp onent of the comprehensive treatment it would have to be formulated in a generalequilibrium and gametheoretic frameworkthat we envision Claims problems and division rules An amount E R has to b e divided among a set N of agents with claims adding up to more than E For each i N let c R denote agent is i claimandc c the vector of claims Initiallywetake N to b e a nite i iN subset of the set of natural numb ers N We sometimes designate by n the N cardinalityof N Altogether a claims problem is a pair c E R R P N suchthat c E Let C denote the class of all problems In Section i An imp ortant step in that direction is taken by Araujo and Pascoa N We denote by R the cartesian pro duct of jN j copies of R indexed by the memb ers of N The sup erscript N also indicates an ob ject p ertaining to the set N Whichinter pretation is the right one should b e clear from the context A summation without explicit b ound should be understo o d to be carried out over all agents We allow the equality P c E for convenience i Vector inequalities x y x y xy we consider situations in which the p opulation of claimants may vary and we generalize the mo del accordingly In our primary application E is the liquidation value of a bankrupt rm the members of N are creditors and c is the claim of creditor i against the i rm A closely related application of the mo del is to estate division a man dies and the debts he leaves b ehind written as the co ordinates of c are found to add up to more than the worth of his estate E How should the estate b e divided Alternatively each c could simply b e an upp er b ound on agent is i consumption without a higher bound necessarily giving him greater rights on the resource N is interpreted as atax assessment problem the When a pair c E C members of N are taxpayers the co ordinates of c are their incomes and they must cover the cost E of a pro ject among themselves The inequality P c E indicates that they can jointly aord the pro ject A dierent i interpretation of c is as the b enet that consumer i derives from the pro ject i Although these various situations can be given the same mathematical description and the principles relevant to their analysis are essentially the same the app eal of each particular prop erty may of course dep end on the application In what follows we mainly think of the resolution of conicting claims Our mo del is indeed a faithful description of the actual situation faced by bankruptcy courts for instance By contrast the issue of taxation is not always sp ecied by rst stating an amount to be collected p erhaps due to the uncertainty p ertaining to the taxpayers incomes Taxation schedules are usually published rst and the amount collected falls wherever it may dep ending