MATH 4321 Game Theory(Spring, 2016)
Tutorial Note XI([email protected])
Couse Review
1. Bargaining games: Bargaining games are cooperative games in which the players bargain
to improve both of their payoffs.
Cooperation here means an agreement as to which combination of strategies each player will
use and the proportion of time that the strategies will be used.
REMARK: The cooperative theory is divided into two classes of problems depending on whether
or not there is a mechanism for transfer of utility from one player to the other.
Transferable utility (TU) or Nontransferable utility (NTU)
2. Pareto-optimal boundary: The Pareto-optimal boundary of the feasible set is the set of
payoff points in which no player can improve his payoff without at least one other player decreasing
her payoff.
There is an incentive for the players to cooperate and try to reach an agreement that will benefit
both players. The result will always be a payoff pair occurring on the Pareto-optimal boundary of
the feasible set.
Pareto optimal: If an agreement is reached in a cooperative game, be it a TU or an NTU game,
it may be expected to be such that no player can be made better off without making at least one
other player worse off. Such an outcome is said to be Pareto optimal.
Definition. A feasible payoff vector, (v1 , v2 ), is said to be Pareto optimal if the only feasible
payoff vector (v10 , v20 ) such that v10 ≥ v1 and v20 ≥ v2 is the vector (v10 , v20 ) = (v1 , v2 ).
REMARK: A win-win situation, a situation that both can improve simultaneously, is not Pareto
optimal.
3. safety point: The status quo payoff point, or safety point, or security point in a two- person
game is the pair of payoffs (u∗ , v ∗ ) that each player can achieve if there is no cooperation between
the players.
4. Nash model with security point: Start with the security status quo point (u∗ , v ∗ ) for a
two-player cooperative game with matrices A. B. This leads to a feasible set of possible negotiated
outcomes depending on the choice of the point (u∗ , v ∗ ).
One convenient choice may be u∗ = value(A) and v ∗ = value(B T ). Given (u∗ , v ∗ ) and feasible
set S, we seek for a negotiated outcome, call it (u, v). Since this point will depend on (u∗ , v ∗ ) and
the set S, so we may write
(u, v) = f (S, u∗ , v ∗ ).
MATH 4321 Game Theory(Spring, 2016)
Tutorial Note XI([email protected])
5. Axioms requirements for the negotiated solution
• Axiom 1. We must have u ≥ u∗ and v ≥ v ∗ . Each player must get at least the status quo
point.
• Axiom 2. The point (u, v) ∈ S, that is, it must be a feasible point.
• Axiom 3. If (u, v) is any point in S, so that u ≥ u and v ≥ v, then it must be the case that
u = u, v = v. In other words, there is no other point in S, where both players receive more.
This is Pareto-optimality.
• Axiom 4. If (u, v) ∈ T ⊂ S and (u, v) = f (S, u∗ , v ∗ ) is the solution to the bargaining
problem with feasible set T , then for the larger feasible set S, either (u, v) = f (S, u∗ , v ∗ )
is the bargaining solution for S, or the actual bargaining solution for S is in S − T . We are
assuming that the security point is the same for T and S. If we enlarge the set of alternatives
from T to S, the new negotiated position cannot be one of the old possibilities in T other
than (u, v).
• Axiom 5. If T is an affine transformation of S, T = aS + b = ϕ(S) and (u, v) = f (S, u∗ , v ∗ )
is the bargaining solution of S with security point (u∗ , v ∗ ) , then (au+b, av+b) = f (T, au∗ +
b, av ∗ + b) is the bargaining solution associated with T and security point (au∗ + b, av ∗ + b).
This says that the solution will not depend on the scale or units used in measuring payoffs.
• Axiom 6. If the game is symmetric with respect to the players, then so is the bargaining
solution. In other words, if (u, v) = f (S, u∗ , v ∗ ) and (i) u∗ = v ∗ , and (ii) (u, v) ∈ S ⇒
(v, u) ∈ S,then u = v. If the players are essentially interchangeable, they should get the
same negotiated payoff.
Analysis of the Axioms
•
•
•
•
•
The 2nd axiom is incontrovertible.
The 1st and 3rd axiom reflects the rationality of the players.
The 6th axiom is a fairness axiom(Symmetry).
The 4th axiom is perhaps the most controversial(Independence of irrelevant alternatives).
The 5th axiom, just reflects the understanding that the utilities of the players are separately
determined only up to change of location and scale(Invariance under change of location and
scale).
6. Theorem Let the set of feasible points for a bargaining game be nonempty and convex, and
let (u∗ , v ∗ ) ∈ S be the security point. Consider the nonlinear programming problem
max g(u, v) := (u − u∗ )(v − v ∗ )
subject to (u, v) ∈ S, u ≥ u∗ , v ≥ v ∗ .
Assume that there is at least one point (u, v) ∈ S with u > u∗ , v > v ∗ . Then there exists one
and only one point (u, v) ∈ S that solves this problem, and this point is the unique solution of the
bargaining problem (u, v) = f (S, u∗ , v ∗ ) that satisfies the axioms 1-6. If, in addition, the game
satisfies the symmetry assumption, then the conclusion of axiom 6 tells us that u = v.
7. Properties: Uniqueness, Pareto optimality
MATH 4321 Game Theory(Spring, 2016)
Tutorial Note XI([email protected])
Problems
1. For the following bimatrix games, draw the NTU and TU feasible sets. What are the Pareto
optimal outcomes?
(0, 4) (3, 2)
(3, 1) (0, 2)
(a)
(b)
(4, 0) (2, 3)
(1, 2) (3, 0)
Solution 1:
The light shaded region is the TU-feasible set. The dark shaded region is the NTU- feasible
region. The NTU-Pareto optimal outcomes are the vectors along the heavy line. The TU-Pareto
outcomes are the upper right lines of slope -1.
MATH 4321 Game Theory(Spring, 2016)
Tutorial Note XI([email protected])
An interesting game
What?s your choices?
There are five prisoners(labeled from 1 to 5). They are arranged to take away beans from a bag
one by one. The total number of beans in the bag is 100 and all the prisoners know this information.
The one who choose the largest number and smallest number beans will be killed. The prisoners
can not discuss and will be told all previous prisoners? choices. So, what?s you choices?