MATH 4321 Game Theory(Spring, 2014)
Tutorial Note XI
Couse Review
1. Theorem of Nash Axioms
There exists a unique function f satisfying the Nash axioms. Moreover, if there exists a point
(u, v) ∈ S such that u > u∗ and v > v ∗ , then f (S, u∗ , v ∗ ) is that point (u, v) of S that maximizes
(u − u∗ )(v − v ∗ ) among points of S such that u ≥ u∗ and v ≥ v ∗ .
2. Zeuthens Principle
Suppose that, at a given stage of the bargaining process, Player 1’s last offer corresponded
to the payoff vector v = (v1 , v2 ) whereas Player 2’s last offer corresponded to the payoff vector
w = (w1 , w2 ) such that v and w belong to the Pareto Optimal boundary of the cooperative feasible
set, with v1 > w1 but v2 < w2 .
If Player 1 accepts his opponent’s last offer w, then he will get payoff w1 . On the other hand, if
he insists on his own last offer v, then he will obtain either payoff v1 or will obtain the disagreement
point payoff d1 (Player 2 sticks to his last offer).
Suppose Player 1 assigns probability p to the latter possibility and assigns probability (1 − p) to
the former possibility. Then, he must conclude that by insisting on his last offer v and by making no
further concession he will get the expected payoff (1 − p)v1 + pd1 , whereas by being conciliatory
and accepting Player 2’s last offer w he will get w1 .
Therefore, Player 1 can rationally stick to his last offer only if (1 − p)v1 + pd1 ≥ w1 i.e. if
1 −w1 )
= r1 .
p ≤ (v
(v1 −d1 )
Similarly, if Player 2 assigns probability q to the hypothesis that Player 1 will stick to his last
offer v (bargaining process ends in disagreement), then Player 2 can rationally stick to his own last
(w2 −v2 )
offer w only if (1 − q)w2 + qd2 ≥ v2 i.e. if q ≤ (w
= r2 .
2 −d2 )
Then, r1 , r2 are the highest probability of a disagreement that Player 1 and Player 2, respectively, would face rather than accept the last offer of the other player.
We will call r1 , r2 the two players’ risk limits expressed as ratio of difference in payoffs.
Zeuthen suggests that the next concession must always come from the player with a smaller risk
limit (except that if the two players’ risk limits are equal then both of them must make concessions
MATH 4321 Game Theory(Spring, 2014)
Tutorial Note XI
to avoid a conflict). We call this suggestion Zeuthen’s Principle.
Theorem: If the two players follow Zeuthens Principle then the next concession will always
made by the player whose last offer is associated with a smaller Nash product (unless both are
associated with equal Nash product, in which case both of them have to make concessions).
Corollary: If the two players act according to Zeuthen’s Principle then they will tend to reach
the Nash solution as their agreement point.
3. The Lambda-Transfer Approach
There is another approach, due to Lloyd Shapley, for solving the NTU-problem that has certain
advantages over the Nash approach.
1. It relates the solution to the corresponding solution to the TU-problem.
2. It avoids the difficult-to-justify fourth axiom.
3. The threat point arises naturally as a function of the bimatrix and does not have to be specified
a priori.
4. It extends to more general problems, but when specialized to the problems with status-quo
point (0,0), it gives the same answer as the Nash solution.
The main difficulty with the NTU-problems is the lack of comparability of the utilities. If we
pretend the utilities are measured in the same units and apply the TU-theory to arrive at a solution,
it may happen that the TU-solution is in the NTU-feasible set. If it happens to be in the NTUfeasible set, the players can use it as the NTU-solution since it can be achieved without any transfer
of utility.
Recall that the utilities are not measured in the same units. Someone might suggest that an
increase of one unit in Player 1’s utility is worth an increase of λ units in Player 2’s utility, where
λ > 0. If that were so, we could analyze the game as follows. If the original bimatrix is (A, B), we
first consider the game with bimatrix (λA, B), solve it for the TU- solution, and then divide Player
1’s payoff by λ to put it back into Player 1’s original units. This is called the λ-transfer game. The
TU-solution to the game with bimatrix (λA, B) is the vector ((σ(λ) + δ(λ))/2, (σ(λ) − δ(λ))/2),
where
σ(λ) = max{λaij + bij } and δ(λ) = Val(λA − B).
ij
The solution to the λ-transfer game is then found by dividing the first coordinate by λ, giving
φ(λ) = (φ1 (λ), φ2 (λ)) = (
σ(λ) + δ(λ) σ(λ) − δ(λ)
,
)
2λ
2
If the point φ(λ) is in the NTU-feasible set, it could, with the justification given earlier, be used as
the NTU-solution.
It turns out that there generally exists a unique value of λ, call it λ∗ , such that φ(λ∗ ) is in the
NTU-feasible set. This φ(λ∗ ) can be used as the NTU-solution. The value of λ∗ is called the
equilibrium exchange rate.
The problem now is to find λ so that φ(λ) is in the NTU-feasible set. In general, this may not
be easy without the assistance of a computer because V al(λA − B) is not a simple function of λ.
However, in one case the problem becomes easy. This occurs for bimatrix games, (A, B), when
the matrices A and −B have saddle points in the same position in the matrix. Such bimatrix games,
when played as NTU games, are said to be fixed threat point games. Whatever be the value of λ
MATH 4321 Game Theory(Spring, 2014)
Tutorial Note XI
in such games, the matrix game, λA − B, which is used to determine the threat point, has a saddle
point at that same position in the matrix. Thus, the threat strategy of the λ-transfer game will not
depend on λ and the threat point is easy to determine.
For NTU-games with a fixed threat point, the λ-transfer solution, φ(λ∗ ), turns out to be the
same as the Nash solution. So in this case we may find the λ-transfer solution by using the method
already described to find the Nash solution.
4. Processes to Lambda-Transfer Approach
1. Plot the NTU feasible set for the bimatrix game [A, B].
2. The resolution point lies in the northeastern boundary of the NTU feasible set.
3. Find the slopes of the line segments of the northeastern boundary. The absolute value of
these slopes are candidates for the exchange rate.
4. If λ is a candidate for exchange rate, solve [λA, B]. The solution, after converting to the scale
of [A, B] is
σ(λ) + δ(λ) σ(λ) − δ(λ)
,
)
φ(λ) = (φ1 (λ), φ2 (λ)) = (
2λ
2
5. If (φ1 (λ), φ2 (λ)) lies in the NTU feasible set then we are done and λ is the equilibrium
exchange rate.
6. If two segments intersect at a vertex v such that on one segment I pays II and on the other
segment II pays I, then the resolution point is v. In this case we cannot determine the equilibrium exchange rate in general. However, for the case of a fixed threat game, we know the
threat point. Then, the line segment joining from the threat point to v is the sharing line. We
can then find the equilibrium exchange rate, λ, in this case. λ = slope of the sharing line.
5. Many-Person TU Games
In many-person cooperative games, there are no restrictions on the agreements that may be
reached among the players. In addition, we assume that all payoffs are measured in the same units
and that there is a transferable utility which allows side payments to be made among the players.
6. Coalitional Form. Characteristic Functions.
Let n ≥ 2 denote the number of players in the game, numbered from 1 to n, and let N denote
the set of players, N = {1, 2, · · · , n}. A coalition, S, is defined to be a subset of N , S ⊆ N , and
the set of all coalitions is denoted by 2N . We also speak of the empty set, Ø, as a coalition, the
empty coalition. The set N is also a coalition, called the grand coalition.
Definition. The coalitional form of an n-person game is given by the pair (N, v), where N =
{1, 2, · · · , n} is the set of players and v is a real- valued function, called the characteristic function
of the game, defined on the set, 2N , of all coalitions (subsets of N ), and satisfying
1. v(Ø) = 0, and
2. (superadditivity) if S and T are disjoint coalitions (S ∩T = Ø), then v(S)+v(T ) ≤ v(S ∪T ).
Compared to the strategic or extensive forms of n-person games, this is a very simple definition.
Naturally, much detail is lost. The quantity v(S) is a real number for each coalition S ⊂ N , which
may be considered as the value, or worth, or power, of coalition S when its members act together
as a unit. Condition (1) says that the empty set has value zero, and (2) says that the value of
two disjoint coalitions is at least as great when they work together as when they work apart. The
assumption of superadditivity is not needed for some of the theory of coalitional games, but as it
seems to be a natural condition, we include it in the definition.
MATH 4321 Game Theory(Spring, 2014)
Tutorial Note XI
7. Relation to Strategic Form
Recall that the strategic form of an n-person game is given by the 2n-tuple, (X1 , X2 , · · · , Xn , u1 , u2 , · · · , un ),
where
1. for i = 1, · · · , n, Xi is the set of pure strategies of Player i, and
2. for i = 1, · · · , n, ui (x1 , · · · , xn ) is the payoff function to Player i, if Player 1 uses x1 ∈
X1 ,Player 2 uses x2 ∈ X2 , ... ,and Player n uses xn ∈ Xn .
Transforming a game from strategic form to coalitional form entails specifying the value, v(S),
for each coalition S ∈ 2N . The usual way to assign a characteristic function to a strategic form
game is to define v(S) for each S ∈ 2N as the value of the 2-person zero-sum game obtained when
the coalition S acts as one player and the complementary coalition, S = N −PS , acts as the other
player, and where the payoff to S is the sum of the payoffs to the players in S: i∈S ui (x1 , · · · , xn ).
Remark: The value, v(S), is the analogue of the safety level. It represents the total amount
that coalition S can guarantee for itself, even if the members of S gang up against it, and have as
their only object to keep the sum of the payoffs to members of S as small as possible. This is a
lower bound to the payoff S should receive because it assumes that the members of S ignore what
possible payoffs they might receive as a result of their actions.
To see that v is a characteristic function, note that Condition (1) holds, since the empty sum is
zero. To see that (2) holds, note that if s is a set of strategies for S that guarantees them v(S), and
t is a set of strategies for T that guarantees them v(T ), then the set of strategies (s, t) guarantees
S ∪ T at least v(S) + v(T ). Perhaps other joint strategies can guarantee even more, so certainly,
v(S ∪ T ) ≥ v(S) + v(T ).
8. Theorem: Given any game in coalition form (N, v), one can find a strategic form game
whose reduction to coalition form has v as its characteristic function.
9. Constant-Sum Games
Definition. A game in coalitional form is said to be constant-sum, if v(S) + v(S) = v(N ) for
all coalitions S ∈ 2N . It is said to be zero-sum if, in addition, v(N ) = 0.
Example: The coalition form of an n-person zero-sum game is a zero-sum game in coalition
form.
10. Imputations and the Core
In cooperative games, it is to the joint benefit of the players to form the grand coalition, N ,
since by superadditivity the amount received, v(N ), is as large as the total amount received by any
disjoint set of coalitions they could form.
It is reasonable to suppose that ”rational” players will agree to form the grand coalition and
receive v(N ). The problem is then to agree on how this amount should be split among the players.
One of the possible properties of an agreement on a fair division, that it be stable in the sense
that no coalition should have the desire and power to upset the agreement.
Such divisions of the total return are called points of the core, a central notion of game theory
in economics.
A payoff vector x = (x1 , x2 , · · · , xn ) means that player i is to receive xi .
The first desirable property of an imputation is that the total amount received by the players
should be v(N ).
Definition. A payoff vector, x = (x1 , x2 , · · · , xn ), is said to be group rational or efficient if
x1 + · · · + xn = v(N ).
No player could be expected to agree to receive less than that player could obtain acting alone.
MATH 4321 Game Theory(Spring, 2014)
Tutorial Note XI
Definition. A payoff vector, x, is said to be individually rational if xi ≥ v({i}) for all i =
1, · · · , n.
Definition. An imputation is a payoff vector that is group rational and individually rational. The
set of imputations may be written
X
{x = (x1 , · · · , xn ) :
xi = v(N ), and xi ≥ v({i}) for all i ∈ N }.
i∈N
The set of imputations is never empty. In fact the set of imputations is exactly the simplex consisting
of the convex hull of the n points obtained by letting xi = v({i}) for all xi except one, which is
then chosen to satisfy x1 + · · · + xn = v(N ).
Essential Games.
P
Definition.
A
game
in
coalitional
form
is
said
to
be
inessential
if
i v({i}) = v(N ), and
P
essential if i v({i}) 6= v(N ).
If a game is inessential, then the unique imputation is x = (v({1}), · · · , v({n})), which may
be considered the ”solution” of the game.
The Core.
If there exists a coalition, S, P
whose total return from x is less than what that coalition can
achieve acting by itself, that is, if i∈S xi < v(S), then there will be a tendency for coalition S to
form and upset the proposed x. Such an imputation has an inherent instability.
P
Definition. An imputation x is said to be unstable through a coalition S if V (S) > i∈S xi . We
say x is unstable if there is a coalition S such that x is unstable through S, and we say x is stable
otherwise.
Definition. The set, C, of stable imputations is called the core,
X
X
C = {x = (x1 , · · · , xn ) :
xi = v(N ) and
xi < v(S), for all S ⊂ N }.
i∈S
i∈S
The core can consist of many points; but the core can also be empty. It may be impossible
to satisfy all the coalitions at the same time. One may take the size of the core as a measure of
stability, or of how likely it is that a negotiated agreement is prone to be upset.
An interesting game
The cannibals
MATH 4321 Game Theory(Spring, 2014)
Tutorial Note XI
A traveler gets lost on a deserted island and finds himself surrounded by a group of n cannibals.
Each cannibal wants to eat the traveler but, as each knows, there is a risk. A cannibal that
attacks and eats the traveler would become tired and defenseless. After he eats, he would become
an easy target for another cannibal (who would also become tired and defenseless after eating).
The cannibals are all hungry, but they cannot trust each other to cooperate. The cannibals
happen to be well versed in game theory, so they will think before making a move.
Does the nearest cannibal, or any cannibal in the group, devour the lost traveler?
The first step is to recognize that the traveler is just a tired and defenseless cannibal. Once you
do that, the problem can be reduced to one just involving cannibals. To solve it now, let’s take small
sample sizes.
Sample Size: n = 1
This is almost a laughably easy problem to solve. If there’s one cannibal, and of course the one
traveler, then the cannibal will obviously eat the traveler, as he has nothing to fear from any other
cannibals.
Sample Size: n = 2
Two hungry cannibals and a defenseless traveler makes for a harder problem. Remember that
we concluded that the traveler is effectively a defenseless cannibal, and by extension a defenseless
cannibal is merely a traveler.
What the cannibals will probably realize that if one of them eats the traveler, they will become
prey for the other cannibal. Therefore, their eating the traveler constitutes their death.
Because of this, neither cannibal will attack and the traveler will be safe.
Sample Size: n = 3
Here’s where the problem gets interesting.
If there are three cannibals and one traveler, this problem can be reduced to one that we have
solved before. Assuming that the cannibals are well versed in game theory, they will figure out that
the first one to attack will reap the benefits.
What do I mean when I say that? If one cannibal eats the traveler, they become a traveler, and
the problem is reduced to n = 2. Therefore, the remaining two cannibals will not attack, and the
first cannibal to attack will be safe.
How do we solve this problem, then? If there are an odd number of cannibals, the cannibals
will attack; Given an even number of cannibals, they will not attack and the traveler will be safe.