For |S| > 1 to make the deposit |S|(w(N) + 2tSε)h({t
S}), each player i ∈ S
has to make a deposit [w(N) + 2tSε]h({t
{i}}) and receive 2tSε revenue in period tS.
Therefore, the deposit can be made by borrowing |S|2tSε, with revenue −|S|2tSε
that can be paid back in period tS + 1. So the intermediate revenue and costs
of the loans cancel out, and the coalition has revenue w(S) at time τ + 1. No additional deposits can be made without borrowing at least w(N), in particular w(N)h({t{i}}) for all i ∈ S is no longer possible without borrowing, and this
is never profitable. So, each player i ∈ S can deposit in only one deposit with non-zero revenue at τ + 1.
The deposits of the form |U|(w(N) + 2tUε)h({t
U}) for all U % S can only be
made by borrowing w(N) · (|U| − |S|) ≥ w(N). Because w(U) − w(N) ≤ 0, this is not profitable.
For each coalition U such that S ∩ U 6= ∅ and S 6⊆ U, U 6⊆ S, a similar argument holds. Because each player i ∈ S ∪ U can deposit in only one deposit with non-zero revenue at τ +1, making both the deposits at tSand tU is impossible
without borrowing at least w(N), which is not profitable.
It is clear that any coalition S can also obtain a total revenue of Pt
k=1v(Sk)
for any partition {S1, S2, . . . , Sk} of S. However, because of superadditivity the
revenue is lower in this case, so the optimal total revenue for the coalition S is w(S). So the deposit game corresponding to this deposit situation is (N, w).
3.7
Reinvestment without debt revisited
We see that the construction of Theorem 3.6.1 depends on loans. The presence of loans ensures that at the relevant periods in time, money is available to the players, although it has to be repaid (possibly with interest) later on. So, it might be beneficial for some coalition to use loans. In the proof of Theorem 3.6.1, the key is that loans are so costly that every player can only afford one loan, which provides him with extra capital for one period only because of the interest payment that follows immediately.
If we do not allow debt, it is unclear whether we still have a characterisation of all non-negative superadditive games. It can be shown for less than four players that the classes of deposit games without reinvestment, with reinvestment, and with reinvestment and debt are equal to the class of non-negative superadditive
games.
Theorem 3.7.1 Let n ≤ 3. Then, every non-negative superadditive game is a deposit game with reinvestment.
Proof: We only have to prove one part of this theorem since Theorem 3.2.3
provides us with the other part. Let (N, w) be a non-negative superadditive game, with n < 3. If N = {1}, it is clear that all non-negative games are deposit games. If N = {1, 2}, take τ = 2 and define the set of all possible terms of a deposit by equation (3.1), and the set of all possible deposits is defined by equation (3.2). We define endowments
m(1) = (1, −1), m(2) = (0, 1). Note that m(N) = (1, 0). We define revenues by
Pτ +1(δ) = w({1}), if δ = (1, −1, 0), w({2}), if δ = (0, 1, −1), w(N), if δ = (1, 0, −1), 0, otherwise,
and Pt(δ) = 0 for all t ≤ τ and all δ ∈ ∆. Let (N, v) be the deposit game
corresponding to this deposit situation. Clearly, players 1 and 2 can obtain their own value individually, and either w({1}) + w({2}) or w(N) when they cooperate. Because the game is superadditive, we have v(N) = w(N) in the deposit game (N, v), hence v = w.
We use a similar construction for N = {1, 2, 3}. Take τ = 3 and define the set of all possible terms of a deposit and the set of all possible deposits by equations (3.1) and (3.2) respectively. The endowments of the players are given by
m(1) = (3, −2, −1), m(2) = (0, 3, −3), m(3) = (0, 1, 2).
3.7. Reinvestment without debt revisited 49
The revenue function is defined by
Pτ +1(δ) = w({1}), if δ = (3, −3, 0, 0), w({2}), if δ = (0, 3, −3, 0), w({3}), if δ = (0, 0, 3, −3), w({1, 2}), if δ = (3, 0, −3, 0), w({1, 3}), if δ = (1, 0, 0, −1), w({2, 3}), if δ = (0, 3, 0, −3), w(N), if δ = (3, 0, 0, −3), 0, otherwise,
and Pt(δ) = 0 for all t ≤ τ and all δ ∈ ∆. Let (N, v) be the deposit game
corresponding to this deposit situation. It is clear that each player i can make exactly w({i}). Any coalition S, consisting of two players, can choose between the sum of its individual players’ choices, and can also obtain w(S). The latter is higher due to superadditivity, and the remaining capital cannot be used to obtain any more revenue. Also coalition N can obtain w(N), and just that. Hence,
w = v.
Whether this is true for more than three players remains an open question. The main difference with the situation where we do allow debt is that it is possible for players to invest their endowments to be able to form coalitions later on. However, these revenues are always available to a player. So if the value of a player is zero, he should not be able to get any revenue at any period in time. So the construction goes awry. We can however show a similar construction if all values are positive. Theorem 3.7.2 Every superadditive game with v(S) > 0 for all S ⊆ N is a deposit game with reinvestment.
Proof: Take a superadditive game (N, w) with w(S) > 0 for all S ⊆ N. We then construct a deposit game with reinvestment that has the same value for any coalition. Define the set of all possible terms of a deposit by equation (3.1), where we take τ = 2|N |. The set of all possible deposits is then defined by equation (3.2). We associate with every period in {1, . . . , τ } a coalition S ⊆ N, such that they are ordered according to the size |S| of the coalition, the smallest being first in
time. One exception to this, is that we take t∅ = τ . We henceforth refer to a
period associated with coalition S as tS ∈ {1, . . . , τ }. Note that there are 2|N | = τ
possible coalitions for the set N of players. We define the endowments of player i ∈ N by mtS(i) = 1, if i ∈ S, mtS−1(i) = 0, −1, if i /∈ S, mtS−1(i) = 1, 0, otherwise,
for all S ⊆ N, where for all i ∈ N we define m0(i) = 0. This implies that
each player has one unit of capital to participate in the period associated with a coalition he takes part in, and zero in all other periods. We define revenues to be positive only in the following situations:
PtS [1 − (2|N|) tSε]h({t {i}}) ! = (2|N|)tSε, ∀i ∈ N, ∀S ∋ i, Pτ +1 [|S| + |S|(2|N|)tSε] h({tS}) ! = w(S) − |S|(2|N|)tSε, ∀S ⊆ N, Pτ +1 h({t{i}}) ! = w({i}), ∀i ∈ N,
For all other t ∈ {1, . . . , τ } and δ ∈ ∆ we set Pt(δ) = 0. We need ε to be small
enough so that all possible revenues remain non-negative, and at most one deposit for one period i possible in each period. Therefore we require
0 < ε < min 1
2, min[w(S)]
(2|N|)τ .
Since we put a bound on the value of ε, we know that 1 − (2|N|)tSε > 1
2, so
only one deposit can be made each period. Each player individually can make at most his own value since intermediate revenues are too low to make another deposit possible. We see here that if a player has value zero, he can still acquire (2|N|)tSε > 0 as revenue.
Only if players in coalition S cooperate, they have enough endowments and intermediate revenues to make a deposit at tS, which has value w(S). Also, since
only one deposit can be made for each individual player, only disjoint coalitions can be formed. Even if every player had a return in t, that would be too little to have enough to deposit in t + 1. Since the game (N, w) is superadditive, they will