A cooperative game with transferable utility, or TU-game, is a pair (N, v) where N = {1, 2, . . . , n} is the set of players, and v : 2N → R is the characteristic
function that assigns a value to each coalition S ⊆ N, representing the payoff to this coalition. By definition, v(∅) = 0. We refer to a cooperative game with transferable utility as ‘a game’, and denote the game by v instead of (N, v) when no confusion arises. For all S ⊆ N, S 6= ∅ the subgame with respect to S is defined as (S, v|S) with v|S(T ) = v(T ) for all T ⊆ S.
A game (N, v) is superadditive if for all S, T ⊆ N such that S ∩ T = ∅ we have v(S ∪ T ) ≥ v(S) + v(T ).
2.2. Games with transferable utility 15
A game (N, v) is called additive if
v(S) =X
i∈S
v({i}),
for all S ⊆ N. The additive game (N, v) corresponding to a vector y ∈ RN is
defined
v(S) =X
i∈S
yi,
for all S ⊆ N. A game (N, v) is convex if for all i ∈ N and all S ⊂ T ⊆ N\{i} it holds
v(S ∪ {i}) − v(S) ≤ v(T ∪ {i}) − v(T ). For a game (N, v), we denote the preimputation set by
P I(N, v) = ( x ∈RN X i∈N xi = v(N) ) .
Any x ∈ P I(N, v) is called efficient. If furthermore x is individually rational, i.e. xi ≥ v({i}) for all i ∈ N, then x is in the imputation set, defined by
I(N, v) = ( x ∈RN X i∈N
xi = v(N), xi ≥ v({i}) for all i ∈ N
) . The core of a cooperative game (N, v) is defined by
C(v) = ( x ∈RN X i∈N xi = v(N), ∀S ⊆ N : X i∈S xi ≥ v(S) ) , and a core allocation is a vector x ∈ C(v).
A cooperative game (N, v) is called balanced if P
S⊆N λ(S)v(S) ≤ v(N) for all
functions λ : 2N → R
+ satisfying
P
S⊆N :i∈Sλ(S) = 1 for all i ∈ N. A game is
called totally balanced if every subgame (S, v|S) is balanced. Bondareva (1963)
and Shapley (1967)derived the following result about the core and balancedness. Theorem 2.2.1 (Bondareva (1963) and Shapley (1967)) Let (N, v) be a cooperative game. Then C(v) 6= ∅ if and only if (N, v) is balanced.
The marginal vector mσ(v) of a game (N, v) corresponding to the permutation
σ is defined by
mσσ(i)(v) = v({σ(1), σ(2), . . . , σ(i)}) − v({σ(1), σ(2), . . . , σ(i − 1)}), for all i ∈ N.
The Shapley value as introduced by Shapley (1953)is defined by Φ(N, v) = 1
|N|! X
σ∈Π(N )
mσ(v), i.e. the average over all marginal vectors.
The excess of a coalition S ⊆ N with respect to the vector x is defined by e(S, x) = v(S) − xS.
This value represents the additional payoff to coalition S compared to x, if S is formed. The excess measures the dissatisfaction of S with the proposed vector x, where lower is better. The aim of the (pre)nucleolus is to minimise the highest excesses. If we consider only efficient allocations, we obtain the prenucleolus of the game (N, v), defined by
ψ(N, v) = { x ∈ P I(N, v)| θ(x) ≤lex θ(y) for every y ∈ P I(N, v)} .
The prenucleolus is always a singleton set. The nucleolus of the game (N, v) is defined by
n(N, v) = { x ∈ I(N, v)| θ(x) ≤lexθ(y) for every y ∈ I(N, v)}
if I(N, v) 6= ∅. The nucleolus is a single point whenever it is defined. Furthermore, the nucleolus is always in the core of the game, if the core is non-empty.
According toSprumont (1990)an allocation scheme (xS)
S⊆N,S6=∅ with xS ∈RS
for all S ⊆ N, S 6= ∅ is called a population monotonic allocation scheme (pmas) for a cooperative game (N, w) if it satisfies efficiency and monotonicity. Here (xS)
S⊆N,S6=∅ satisfies efficiency if
X
i∈S
xSi = w(S),
for all S ⊆ N, S 6= ∅, and monotonicity if for each T ⊃ S and i ∈ S it holds that xTi ≥ xSi .
2.2. Games with transferable utility 17
If (xS)
S⊆N,S6=∅ is a pmas, then for all S ⊆ N, S 6= ∅, xS belongs to the core of the
subgame (N, v|S) restricted to S and hence (N, v) is totally balanced.
An extension of a game with transferable utility is a fuzzy game with trans- ferable utility (N, r), where N is the player set and r : [0, 1]N → R is the char-
acteristic function. Players are assumed to be infinitely divisible, hence coalitions consist of fractional players, i.e. s ∈ [0, 1]N. By definition r(0) = 0. We denote
as =Pi∈Nsi · ai.
Similar to the core of a game, the fuzzy core of a fuzzy game r is defined by F Core(r) = {x ∈RN|x
s≥ r(s) for all s ∈ [0, 1]N,
X
i∈N
xi = r(1, 1, . . . , 1)}.
Aubin (1979) shows that if the function r is convex, then the fuzzy core is non- empty, convex and compact. Furthermore, if r is differentiable at 1, then the fuzzy core is a singleton set.
Chapter 3
Deposit games with reinvestment
The Yen Buddhists are the richest religious sect in the universe. They hold that the accumulation of money is a great evil and a burden to the soul. They therefore, regardless of personal hazard, see it as their unpleasant duty to acquire as much as possible in order to reduce the risk to innocent people.Terry Pratchett Witches Abroad
3.1
Introduction
A deposit problem is a decision problem where an individual has a certain amount of capital at his disposal to deposit at a bank. This individual aims to maximise the return on his investment. When we allow for cooperation between individuals by means of joint investment, they can increase their joint return. However, this gives rise to the additional question of how to allocate the proceeds among the individuals. In this chapter, we analyse cooperation in deposit situations and in particular, we explore whether an allocation exists such that all players will want to cooperate and form the grand coalition.
Deposit games have previously been studied byBorm, De Waegenaere, Rafels, Suijs, Tijs, and Timmer (2001), Tan (2000) and De Waegenaere, Suijs, and Tijs (2005). Borm et al. (2001) define a deposit as a fixed amount of capital at a bank for a certain amount of time, and use a revenue function that describes the
revenue of a deposit in a fixed end period, after which no more deposits can be made. In particular, they do not allow for reinvestment of intermediate revenue.
Tan (2000)extends this approach and adds the possibility of borrowing.
The approach of De Waegenaere et al. (2005) is more general. They allow reinvestment and a broader range of investment products than deposits. Among other things, the money invested in a single investment product is allowed to change every period, rather than being a fixed amount of capital. The drawback of this approach is that because of its generality, it is harder to draw firm conclusions.
Lemaire (1983)andIzquierdo and Rafels (1996)analyse related issues. Lemaire (1983) gives an overview of the use of game theory in financial issues. Izquierdo and Rafels (1996) analyse games that are related to capital dependent deposit games.
In this chapter, which is based onVan Gulick, Borm, De Waegenaere, and Hen- drickx (2010), we take the approach that reinvestment should be possible, while still allowing for a particular form of the revenue function. We start by modelling the decisions of individuals during a discrete and finite number of periods of time. Any deposit will lead to a non-negative revenue. This non-negativity condition is justified because, by the nature of deposits, it is always possible to privately save any amount of money at no cost. There will be no default risk involved in any deposit.
We allow coalitions of individuals to deposit money jointly by cooperating. Clearly, the revenue of a coalition will never be lower than the sum of all individual revenues. We are interested in finding a core element, i.e. we want an allocation for the grand coalition such that no coalition has any incentive to split off.
Particular forms of the revenue function are explored by introducing two sub- classes: term dependent deposit games and capital dependent deposit games. These classes are also considered by Borm et al. (2001). Each of the classes fo- cusses on one of the two main decisions an investor faces when depositing. These main decisions are the amount of capital to deposit and the length of time over which to deposit capital. First we focus on situations where the prime decision is on the length of time; the influence of the amount of capital is assumed to be constant. These situations are called term dependent. We show that the associate term dependent deposit games are totally balanced. The natural counterpart of term dependent deposit games are capital dependent deposit games; the prime