Bankruptcy Problem from the Talmudsyha/Bankruptcy.pdfBankruptcy Problem from the Talmud Introduction...

Bankruptcy Problem from the Talmud


HeeJoon Tak

May 22, 2013


Original paper

Game Theoretic Analysis of a Bankruptcy Problem from theTalmud

Robert J. Aumann and Michael Maschler

Journal of Economic Theory 36. 195-213 (1985)


Outline

1 Introduction

2 Consistent Solution

3 Connection with coalitional game


Introduction

Outline

1 Introduction




Introduction

Bankruptcy Problem

A man dies, leaving debts d1, d2, · · · , dn totalling more thanhis estate E . How should the estate be divied among thecreditiors?


Introduction


Debt100 200 300

100 1003

1003

1003

Estate 200 50 75 75

300 50 100 150


Consistent Solution

Outline

1 Introduction




Consistent Solution

The Contested Garment

Two holds a garment, one claims it all, the other claims half.Then the one is awarded 3/4, the other 1/4.

The lesser claimant concedes half the garment to the greaterone. The remaining half is divided equally.


Consistent Solution

Two person bankruptcy problem

Estate is E and claims d1, d2.

The amount that i concedes to the other claimant j is(E − di )+. Therefore the solution is given by

xi =E − (E − d1)+ − (E − d2)+

2+ (E − dj)+.

We will say that this division is prescibed by the CG(contestedgarment) principle.


Consistent Solution

Consistency

A bankruptcy problem is defined as a pair (E ; d), whered = (d1, d2, · · · , dn), 0 ≤ d1 ≤ d2 ≤ · · · ≤ dn and0 ≤ E ≤ D := d1 + · · · dn.

A solution to such a problem is an n − tuple x = (x1, · · · , xn)of real numbers with

x1 + x2 + · · ·+ xn = E .


Consistent Solution

Consistency

A solution is called CG-consistent or consistent if for all i 6= j ,the division of xi + xj prescribed by the CG-principle for claimsdi , dj is (xi , xj).

Remark

The bankruptcy problem from the Talmud are consistent solution.


Consistent Solution

Consistent solution of bankruptcy problem

Theorem 1

Each bankruptcy problem has the unique consistent solution.


Consistent Solution

Uniqueness of consistent solution

Let x , y be two consistent solutions of given bankruptcy problem.We can find i 6= j such that

yi > xi , yj < xj , yi + yj ≥ xi + xj .

Since CG principle has monotonicity with respect to estate,yj ≥ xj . It is contradiction.


Consistent Solution

Existence of consistent solution

Case1) E ≤ D/2.

When E is small, all n claimants divide it equally.

This continues untill 1 has received d1/2. For the time beingshe then stops, and each additional dollar is divided equallybetween the remaining n − 1 claimants.


Consistent Solution


This, in turn, continues until 2 has received d2/2, at whichpoint she stops receving payment for the time being, and eachadditional dollar is divided equally between the remainingn − 2 claimants.


Consistent Solution


Case2) E > D/2.

The process is the mirror image of the above. Instead ofthinking i ’s award xi , one thinks in terms of her loss di − xi .

When D − E is small, the tatal loss is shared equally betweenall creditors.

This continues untill 1 has lost d1/2. For the time being shethen stops losing and each additional loss is divided equallybetween the remaining n − 1 claimants.


Consistent Solution


Remarks

1 Above gives the consistent solution.

2 Let us denote f (E ; d) be the unique consistent solution ofbankruptcy problem (E ; d). Then the following duality holds;

f (E ; d) = d − f (D − E ; d).


Connection with coalitional game

Outline

1 Introduction





Definition of bankruptcy game

Definition

We define the bankruptcy game vE ;d corresponding to thebankruptcy problem (E ; d) by

vE ;d := (E − d(N \ S))+.

Remark

vE ;d is a coalition game with superadditivity.

vE ;d(S): Accepting either nothing, or what is left of the estate Eafter each menber i of the coalition N \ S is paid his completeclaim di .



Main theorem

Theorem 2

The consistent solution of a bankruptcy problem is the nucleolus ofthe corresponding game.

From now on, let v be a game, S a coalition and x a payoff vector.



The reduced game

Definition

The reduced game vS ,x is defined on the player space S as

vS ,x(T ) =

{x(T ) if T = ∅ or T = S

maxQ⊂N\S{v(Q ∪ T )− x(Q)} if ∅ ( T ( S

In the reduced game, the players of S consider how to divide thetatal amount assigned to them by x under the assumption that theplayers i outside S get exactly xi .



The reduced game

Lemma 3

Let x be a solution of the bankruptcy problem (E ; d) such that0 ≤ xi ≤ di for all i . Then for any coalition S the following holds;

vS ,xE ;d = vx(S);d |S .

proof. Set v := vE ;d , vS := vS ,xE ;d . First let ∅ ( T ( S and let the

maximum in the definition of vS(T ) be attained at Q.



The reduced game

vS(T ) = v(T ∪ Q)− x(Q)

= [E − d(N \ (Q ∪ T ))]+ − x(Q)+

≤ [x(N)− d(N \ (Q ∪ T ))− x(Q)]+

= [x(S)− d(S \ T )− (d − x)(N \ (S ∪ Q))]+

≤ (x(S)− d(S \ T ))+

Setting Q = N \ S

vS(T ) ≥ v(T ∪ (N \ S))− x(N \ S)

≥ [E − d(N \ (T ∪ (N \ S)))]+ − (x(N)− x(S))

≥ (E − d(S \ T ))− (E − x(S)) = x(S)− d(S \ T ).



The reduced game

and setting Q = ∅

vS(T ) ≥ v(T ∪ ∅)− x(∅) = v(T ) = (E − d(N \ T ))+ ≥ 0.

Therefore vS(T ) = (x(S)− d(S \ T ))+ = vx(S);d |S(T ).



Kernel and pre-kernel

Let v be a game. For each payoff vector x and players i , j , define

Sij(x) = max{v(S)− x(S)|S contains i but not j}.

Definition

The pre-kernel of v is the set of all payoff vectors x withx(N) = v(N) and Sij(x) = Sji (x) for all i 6= j .

Definition

The kernel of v is the set of all payoff vectors x with x(N) = v(N),xi ≥ v(i) for all i and for all i 6= j

Sij(x) > Sji (x) implies xi = v(j).



Standard solution of 2 person game

For a given 2 person game v , the payoff vector

xi =v(12)− v(1)− v(2)

2+ v(i)

is called standard solution of v .

Above definition is equivalent with x1 + x2 = v(12) andx1 − x2 = v(1)− v(2).

Remark

The nucleouls and kernel, the pre-kernel and the Shapley value of 2person game all coincide with its standard solution.



Pre-kernel and standard solution

Lemma 4

Let x be in the pre-kernel of a game v and let S be a coalitionwith exactly two person. Then x |S is the standard solution of vS ,x .

proof) Let S = {i , j} Then

Sij(x) = maxQ⊂N\S

(v(Q ∪ i)− x(Q ∪ i)) = maxQ⊂N\S

(v(Q ∪ i)− x(Q))− xi

= vS ,x(i)− xi .

Similarly, we know that Sji (x) = vS ,x(j)− xj . Since x is in thepre-kernel Sij(x) = Sji (x). This implies

xi − xj = vS ,x(i)− vS ,x(j)

also by definition of reduced game

xi + xj = x(i , j) = vS ,x(i , j).



Kernel and the consistent solution of bankrutpcy problem

Lemma 5

The contested garment solution of 2 person bankruptcy problem isthe standard solution of the corresponding game.

Proof.

Directly from Lemma 4 and definition of bankruptcy game and thestandard solution.



Kernel and the consistent solution of bankrutpcy problem

Proposition 6

The kernel of a bankruptcy game vE ;d consists of a single point,namely the consistent solution of the problem (E ; d).

Proof) Set v := vE ;d . v is superadditive, hence 0-monotonic, i.e.S ⊂ T implies v(S) +

∑i∈T\S v(i) ≤ v(T ). In 0-monotonic game,

the kernel coincides with the pre-kernel. Hence x is in thepre-kernel of v . Let S be an arbitrary 2 person coalition. Bylemma 4, x |S is the standard solution of vS ,x and hence, ofvx(S);d |S . Therefore x |S is the CG-solution of (x(S); d |S), but thatmeans x is the consistent solution of (E ; d).



Proof of main theorem

Theorem

The consistent solution of a bankruptcy problem is the nucleolus ofthe corresponding game.

Proof.

Since the nucleolus is always in the kernel, by proposition 6, itcoincides with the consistent solution of the bankruptcyproblem.

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