Summary Games and Economic Behavior (Games) | EOR Year 3 Tilburg University

Games and Economic Behaviour
Max Batstra
March 29, 2026




1

, 1. TU Games
Setup Cooperative TU-games
• N players
• Coalition S is a set of players that cooperate (2N possible coalitions)
• Characteristic function v : 2N → R, that assigns value to each player in coalition S ⊆ N


Definition 1. [Simple Game]
A monotonic game (N, v) is called simple if:
• v(S) ∈ {0, 1} for all S ∈ 2N a
• v(N ) = 1


Definition 2. [Superadditivity]
A T U -game (N, v) is called superadditive if

v(S ∪ T ) ≥ v(S) + v(T ) for all S, T ∈ 2N with S ∩ T = ∅


Definition 3. [Additive]
A game (N, v) is called additive if

v(S ∪ T ) = v(S) + v(T ) , for all S, T ∈ 2N with S ∩ T = ∅


Definition 4. [S-equivalent]
Two T U -games (N, v) and (N, w) are called S-equivalent if
X
∃k > 0, a ∈ RN : w(S) = k · v(S) + ai
i∈S

Example: See exercise 1.3


Definition 5. [Zero-normalized]
A game v ∈ T U N is called zero-normalized if:

v({i}) = 0 for all i ∈ N


Definition 6. [Unanimity Game]
The for each T ∈ 2N \{∅} unanimity game is defined:
(
1 if T ⊆ S
uT (S) =
0 otherwise



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, 2. The Core
Consider a game v ∈ T U N . In a cooperative game, players are negotiating about the formation of the grand
coalition N , which is based on the allocation of v(N ). Feasible allocations of v(N) to the players in the grand
coalition are called imputations


Definition 7. [Imputation]
Consider a game v ∈ T U N , an allocation x ∈ RN is called an imputation if x satisfies:
X
1. Efficiency: xi = v(N )
i∈N

2. Individual rationality: xi ≥ v({i}) for all i ∈ N
and the set of all imputations of the game v ∈ T U N is denoted by I(v)


Definition 8. [The Core]
The core C(v) of a game v ∈ T U N is defined by:
( )
X X
C(v) = x ∈ RN xi = v(N ), xi ≥ v(S) for all S ∈ 2N
i∈N i∈S

Note: The core satisfies relative invariance w.r.t. S-equivalence, i.e. if w = kv + a (k > 0, a ∈ RN ,
additive), then x ∈ C(v) =⇒ kx + a ∈ C(w) and reversely.

Example Consider the game Then finding the core, is equivalent to solving the linear system of equations,

S ∅ {1} {2} {3} {1, 2} {1, 3} {2, 3} N
v(S) 0 0 0 0 10 10 30 35

satisfying:



 x1 + x2 + x3 = 35 
∗
x1 + x2 ≥ 10 0 ≤ x1 ≤ 5


 
x1 + x3 ≥ 10 =⇒ 0 ≤ x∗2 ≤ 25
0 ≤ x∗3 ≤ 25
 
x2 + x3 ≥ 30





x , x , x ≥ 0
1 2 3

From this, the core follows:

C(v) = Conv {(5, 25, 5); (5, 5, 25); (0, 25, 10); (0, 10, 25)}

Finding the core is done by finding the bounds on x, and then taking the convex hull between, for each xi ,
all permutations where
P each xi is set to its upperbound, and the rest of the x−i take values such that the
bounds are met, and i∈S xi = v(N )


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,Definition 9. [Linear Production (LP) - problem]
A linear production (LP) - problem can be described by a tuple (R, P, A, b, c) and is defined:

max cT x s.t. Ax ≤ b with x ∈ RP , x ≥ 0

where:
• R a finite set of resources
• P a finite set of products
• A a linear technology matrix, where Arp is the number of units of resource r ∈ R to produce one unit
of product p ∈ P
• b ∈ RR supply bundle of resources
• c ∈ RP market prices per unit of product

To evaluate LP -processes, we consider all possible LP -problems for all coalitions S ∈ 2N , which leads to the
corresponding LP-game vL ∈ T U N defined by:
X Duality X
vL (S) = max{x⊤ c|Ax ≤ bi , x ≥ 0} ⇐⇒ vL (S) = min{y ⊤ bi | y ⊤ A ≥ c⊤ , y ≥ 0}
i∈S i∈S



Algorithm Calculation of LP-game + finding corresponding Core
1. Find the l corner points of F ∗ of dual problem, i.e. {y ∗ | y ⊤ A = c⊤ , y ⊤ ≥ 0⊤ } (use drawing)
2. create matrix

S y1∗ ... yl∗ vL (S)
∗ ⊤ ∗ ⊤
S1 (y1 ) b1 ... (yl ) b1 min{values in this row}
..
.
(y1∗ )⊤ (yl∗ )⊤
P P
N i∈N bi ... i∈N bi min{values in this row}


3. In 2. we’ve found the game vL , P
C(vL ) can be found
Pusing the usual algorithm for finding the core of a
T U -game v: C(vL ) = {x ∈ RN | i∈N xi = vL (N ), i∈S xi ≥ vL (S) for all S ∈ 2N }


Definition 10. [OwenSet] 

For an LP -process L = N, R, P, A, bi i∈N
, c we define the Owen set Owen(L) by:
( )
X
⊤ N ∗ ⊤ i
Owen(L) = (y bi )i∈N ∈ R y ∈ F , vL (N ) = y b
i∈N

Here F ∗ is the feasible region from the dual problem of the LP -problem. Elements of te Owen set are called
Owen vectors.


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, In order to find the Owen set for a LP -process, we use steps 1 and 2 of the previous algorithm, but only need
to find vL (N ) in step 2. This is because we only need to know vL (N ), whereas for the core of the problem
we need to know vL (S) for all S ∈ 2N .


Theorem 1. For any LP -process L, we have: Owen(L) ⊆ C(vL )




3. Strategic Games and Nash Equilibria
Definition 11. [Strategic Game]
A strategic game G with a finite set N of players is given by:

G = {(Xi , πi )i∈N }

Here, for each i ∈ N :

- Xi is the strategy set of player i (set of actions player i can make)
Q
- πi : j∈N Xj −→ R is the payoff function of player i which assigns a value to player i for all possible
game outcomes.


Definition 12. [Nash Equilibrium]
A strategy combination x∗ = (x∗j )j∈N is called a Nash equilibrium if:

πi (x∗ ) ≥ πi (xi , x∗−i ) for all xi ∈ Xi

The set of all Nash equilibria of G is denoted by E(G)
Interpretation: For each player, there are no profitable deviations from x∗


Definition 13. [Dominant Strategy]
A strategy x∗i ∈ Xi is called dominant for player i if:

πi (x∗i , x−i ) ≥ πi (xi , x−i ) for all xi ∈ Xi and x−i ∈ X−i

Interpretation: Against all possible strategies x−i from other players, there are no profitable deviations
from x∗i for player i


Definition 14. [Reaction Curve]
Consider a game G, then the reaction curve of player i describes what player i should optimally play as a
function of the other players strategy.




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