Survey of bankruptcy problems
with non-standard features
Vito Fragnelli
Università del Piemonte Orientale
[email protected]
Joint work with:
Stefano Gagliardo
[email protected]
Fabio Gastaldi
[email protected]
Compromesso di
Lussemburgo - 1966
Seminario POLIS di Inverno
30 Gennaio 2015
Survey of bankruptcy problems with non-standard features
Summary
Standard bankruptcy problems
Axioms
Non-standard bankruptcy problems
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Survey of bankruptcy problems with non-standard features
3
Standard bankruptcy problems
A bankruptcy problems BP arises when an agent has several (monetary) debts with other
agents and her/his (monetary) availability is not enough for cover all of them
The creditors have the same rights on the available estate
Similar situations are the allocation of a scarce resource, or the collection of taxes
Two very good surveys may be found in Thomson (2003 and 2015)
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Survey of bankruptcy problems with non-standard features
Formally
A BP is a triple
B = (N, E, c)
where N = {1, ..., n} is the set of claimants, E ∈ R≥ is the estate and c = (c1 , ..., cn) ∈ Rn≥
P
is the vector of claims, with E ≤ i∈N ci = C
O’Neill (1982), Aumann, Maschler (1985), Curiel, Pederzoli, Tijs (1987)
A solution is a vector x = (x1 , ..., xn) ∈ Rn s.t.
0 ≤ xi ≤ ci , i ∈ N
P
i∈N xi = E
(rationality)
(efficiency)
A rule is a map f that associates to each BP a solution
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Survey of bankruptcy problems with non-standard features
Basic rules
Proportional
P ROP (N, E, c)i = ci
E
,i ∈ N
C
Constrained Equal Awards
CEA(N, E, c)i = min{ci, α}, i ∈ N
P
where α ∈ R≥ is s.t. i∈N CEA(N, E, c)i = E
Constrained Equal Losses
CEL(N, E, c)i = max{ci − β, 0}, i ∈ N
where β ∈ R≥ is s.t.
Talmud
P
CEL(N, E, c)i = E
(
CEA(N, c/2, E)i
if E ≤ C/2
T AL(N, E, c)i = ci
+ CEL(N, c/2, E − C/2)i if E > C/2
2
Herrero, Villar (2001)
i∈N
Survey of bankruptcy problems with non-standard features
Axioms
How to select the “best” solution?
Each agent prefers the solution that allows to obtain the largest amount
GOOD PROPERTIES −→ AXIOMATIC CHARACTERIZATIONS
In the literature there exist more than 100 properties, perhaps more than 200
• Strong properties: Order preservation, Equal treatment of equals, Monotonicity, etc
• Context properties: Claim truncation, Merging, Splitting, etc
• Ad hoc properties: Exemption, Full compensation, etc
Claim truncation → game theoretical rules (Curiel, Pederzoli, Tijs, 1987)
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Survey of bankruptcy problems with non-standard features
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Non-standard bankruptcy problems
MORE DATA
Each agent is represented by a unique datum, the claim
• Weights. A non-negative real number is associated to each agent, and this influence the
rules
Casas-M´endez, Fragnelli, Garc´ıa-Jurado (2011)
• Minimal rights and bounds. It is possible to introduce the minimal right of an agent as the
amount of the estate nobody else claims, or bounds on the amount that has to be assigned
to the agent (What happens when the sum of the lower bounds is strictly larger than the
estate?)
Pulido, Sanchez-Soriano, Llorca (2002) and Moreno-Ternero, Villar (2004)
• Multiclaims. Each agent may have more than one claim for the unique estate (How to
aggregate the the claims of each agent? Conversely, how to split the estate?)
Calleja, Borm, Hendrickx (2005) and Hinojosa, M´armol, S´anchez (2012)
Survey of bankruptcy problems with non-standard features
DIFFERENT RIGHTS
• Types (Young, 1998)
• Priority (Moulin, 2000)
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Survey of bankruptcy problems with non-standard features
REPRESENTATION
• Communicating vessels (Kaminski, 2000)
Extremely easy to adapt to different situations, thanks to shapes, sections, heights, levels
E
A
A
A
A
c2
A
c4 /2
c3
m3
c1 maximal
priority
c4 /2 m4
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Survey of bankruptcy problems with non-standard features
• Network
– Minimum cost flow problem (Branzei, Ferrari, Fragnelli, Tijs, 2006)
'
$
E/E
source
&
'
%
-
&
– Flow problem (Bjørndal, J¨ornsten, 2010)
$
'
$
%
&
%
m1/c1 /k1(x1 ) ..
sink
mn/cn /kn(xn)-
– In a multiple BP , the network describes the connections with each source
Ilkılı¸c, Kayı (2014) and Moulin, Sethuraman (2013)
Survey of bankruptcy problems with non-standard features
DIFFERENT UTILITY FUNCTIONS
Carpente, Casas, Gozalvez, Llorca, Pulido, Sanchez-Soriano (2008)
A PRIORI UNIONS
Borm, Carpente, Casas-M´endez, Hendrickx (2005)
SURPLUS
Moulin (1987) and Herrero, Maschler, Villar (1999)
INTEGER ESTATE
Wait some minutes ...
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Survey of bankruptcy problems with non-standard features
References
Aumann RJ, Maschler M (1985) Game theoretic analysis of a bankruptcy problem from the Talmud. Journal of Economic Theory 36, 195-213
Bjørndal E, J¨
ornsten K (2010) Flow sharing and bankruptcy games. International Journal of Game Theory 39, 11-28
Borm P, Carpente L, Casas-M´endez B, Hendrickx R (2005) The constrained equal awards rule for bankruptcy problems with a Priori Unions. Annals of
Operations Research 137 , 211-227
Branzei R, Ferrari G, Fragnelli V, Tijs S (2008) A flow approach to bankruptcy problems. AUCO Czech Economic Review 2, 146-153
Calleja P, Borm P, Hendrickx R (2005) Multi-allocation situations. European Journal of Operations Research 164, 730-747
Carpente L, Casas B, Gozalvez J, Llorca N, Pulido M, Sanchez-Soriano J (2008) How to divide a cake when people have different metabolism? Mathematical
Methods of Operations Research 78, 361-371
Casas-M´endez B, Fragnelli V, Garc´ıa-Jurado I (2011) Weighted bankruptcy rules and the museum pass problem. European Journal of Operational Research
215, 161-168.
Curiel I, Maschler M, Tijs S (1987) Bankruptcy games, Zeitschrift f¨
ur Operations Research 31, 143-159
Herrero C, Maschler M, Villar A (1999) Individual rights and collective responsibility: the rightsegalitarian solution. Mathematical Social Sciences 37, 59-77
Herrero C, Villar A (2001) The Three Musketeers: Four Classical Solutions to Bankruptcy Problems. Mathematical Social Sciences 42, 307-328
Hinojosa MA, M´armol AM, S´anchez F (2012) A consistent Talmud rule for division problems with multiple references. TOP 20, 661-678
Ilkılı¸c R, Kayı C (2014) Allocation rules on networks. Social Choice and Welfare 43, 877-892
Kaminski MM (2000) Hydraulic Rationing. Mathematical Social Sciences 40, 131-155
Moreno-Ternero J, Villar A (2004) The Talmud rule and the securement of agents awards. Mathematical Social Sciences 47, 245-257
Moulin H (1987) Equal or Proportional Division of a Surplus, and Other Methods. International Journal of Game Theory 16, 161-186
Moulin H (2000) Priority Rules and Other Asymmetric Rationing Methods. Econometrica 68, 643-684
Moulin H, Sethuraman J (2013) The bipartite rationing problem. Operations Research 61, 1087-1100
O’Neill B (1982) A Problem of Rights Arbitration from the Talmud. Mathematical Social Sciences 2, 345-371
Pulido M, Sanchez-Soriano J, Llorca N (2002) Game theory techniques for university management: an extended bankruptcy model. Annals of Operations
Research 109, 129-142
Thomson W (2003) Axiomatic and Game-Theoretic Analysis of Bankruptcy and Taxation Problems: a Survey. Mathematical Social Sciences 45, 249-297
Thomson W (2015) Axiomatic and Game-Theoretic Analysis of Bankruptcy and Taxation Problems: an Update. to appear in Mathematical Social Sciences
Young HP (1994) Equity, in Theory and Practice. Princeton University Press
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