Game Theory Exam 2018 Cheat
Sheet
Non- Cooperative Game Theory
Players: N= 1,2,.., n
Actions/strategies: Each player chooses sifrom his own
finite strategy set: Sifor each i∈Nresulting in a tuple that
describes strategy combination: s= (s1, ..., sn)∈(Si)i∈N
Payoff outcome: ui=ui(s) for some chosen strategy
best-response: Player i’s best-response to the strategies s−i
played by all others is the strategy s∗
i∈Sisuch that
ui(s∗
i, si)≥ui(s‘
i, s−i)∀s‘
i∈Siands‘
i6=s∗
i
Pure - strategy (Nash Equilibrium): All strategies are
mutual best responses:
ui(s∗
i, si)≥ui(s‘
i, s−i)∀s‘
i∈Siands‘
i6=s∗
i
Cooperative game
Population of players: N= 1,2, ..., n
Colations: C⊂Nform in the population and become
players results in a coalition structure: ρ={C1, C2, ..., Ck}
Payoffs: Φ = {Φ1,..., Φn}and we need a sharing rule for for
the individual player resulting in: Φi= Φ(ρ, ”sharing rul e”)
characteristic function form (CFG): The Game is defined
by 2-tuple G(v, N) where Characteristic function:
v: 2N→Rwhere 2Nare all possible coalitions.
transfer of utils: occures when we share the value of the
characteristic function among the participants of the coalition.
and feasibility is then when: Pi∈CΦi≤v(C)
Superadditivity: If two coalitions C, S are disjoint then
v(C) + v(S)≤v(C∪S)
The Core of a superadditive G(v, N) consists of all outcomes
where the grand coalition forms and payoff allocations
Φ = (Φ1, ..., Φn) are:
1. Pareto efficient: Pi∈NΦi=v(N) so whole value is out
2. Unblockable: ∀C⊂N, Pi∈CΦi≥v(C) so payout for
each individual is bigger then if it would act
alone(individual rational) or in a
sub-coalition(coalitional rational) of N.
nonempty core: if and only if the game is balanced.
balancedness:
1. Balancing weight: attached to each Coalition C:
α(C)∈[0,1]
2. Balanced family: A set of balancing weights is balanced
family if Pi∈Cα(Ci)=1
3. Balancedness in superadditivity: requires that for all
balanced families: v(N)≥PC∈2Nα(C)v(C)
Shapley value
Pays each player average marginal contributions. Marginal
contributions: For any S: i∈S, think of marginal
contributions as : M Ci(S) = v(S)−v(S\i)
Given some G(v, N), an acceptable allo cation/value x∗(v)
should satisfy:
1. Efficiency: Pi∈Nx∗
i(v) = v(N)
2. Symmetry: if for any two players i and j,
v(S∪i) = v(S∪j) so player i,j same influence on value
v(S). then x∗
i(v) = x∗
j(v)
3. Dummy player if for any i, v(S∪i) = v(S) for all S
then x∗
i(v)=0
4. Additivity if u,v are two characteristic functions then
x∗(v+u) = x∗(v) + x∗(u)
Shapley function:
Φi(v) = PS∈N,i∈S=(|S|−1)!(n−|S|)!
n!(v(S)−v(S\i)). It pays
average marginal contributions.
Non transferable-utility cooperative game: Game:
G(v, N )outcome: partition ρ=C1, C2, ..., Ckimplies directly
a payoff allocation. e.g. only coalitions of pairs. Deferred
acceptance: For any marriage problem, one can make all
matchings stable using the deferred acceptance algorithm.
1 Initialize) all mi∈Mand all wi∈Ware single.
2 Engage) Each single man m∈Mproposes to his preferred
woman wto whom he has not yet proposed. a) If w is single,
she will become engaged with her preferred proposer. b) Else
w is already engaged with m’: if w prefers prop oser m over m’
she becomes engaged with m and m’ becomes single. If not
(m’, w) remain engaged. c) All prop osers wo do not become
engaged remain single. 3 Repeat) If there exists a single man
after Engage repeat Engage. Else Terminate 4 Terminate)
Marry all engagements
Preferences and utility
A binary relation (weakly prefers), (prefers), ∼
(indifferent) on a set X is a non-empty subset P⊂X×X. We
write xyiff (x, y)∈P
Assumptions on preferences:
1. Completeness: ∀x, y ∈X:xyory xorboth so we
have some preference for any element to any other in
the set.
2. Transitivity: ∀x,y, z ∈X: if xyand yzthen
xz
3. Continuity:
W(x) = y∈X:xy, B(X) = y∈X:yxso we
have once all below x and once all above x then
continuity tells us that we do not have some kind of big
gap or between these. ∀x∈X:B(x) and W(x) are
closed sets.(including their boundary points)
4. Independence of irrelevant alternatives: ∀x, y, z ∈X:
xy⇒(1 −λ)x+λz (1 −λ)y+λz =x+zy+z
A utility function for a binary relation on a set X is a
function u:X→Rsuch that
u(x)≥u(y)⇔xy
so give the preference an actual value and still preserving the
preference. There exists such a utility function for each
complete, transitive, positively measurable and continuous
preference on any closed or countable set.
Ordinal utility function: difference between u(x) and u(y)
is meaningless. Only u(x)≥u(y) is meaningful.
Cardinal utility function: A utility function where
differences between u(x) and u(y) are meaningful as they
reflect the intensity of preferences. (invariant to positive affine
transformations)
Utils: An even stronger statement would be that there is a
fundamental measure of utility. say one ”util”. It is not
invariant to any transformation.
Lottery Let X be a set of outcomes then a lottery on X
means nothing but a probability distribution on X. The set of
all lotteries on X is usually denoted by ∆(X). E.g.
X= (x1, ..., xK) then a lottery is represented by (p1,..., pK)
and they should sum to one.
Decision problem under risk: Is then when the decision
maker has to choose a lottery from a Set of available lotteries:
C⊆∆(X)
St. Petersburg Paradox: A rational decider would prefer
lotteries with higher expected payoff. E[l] ¿ E[l’] but this leads
to a paradox when using infinity expected values.
Expected utility maximization: Was introduced to solve
St. Petersburg Paradox. So instead of weighting lotteries
directly on their payoff we weight them on their utility
function.
Utility function on lotteries: A preference relation on
∆(X) is sait to be representable by a utility function U
whenever for every lotteries p:= (p1,..., pk) and
p0:= (p0
1, ..., p0
k), pp0only when U(p)≥U(p0)
Bernouilli function is the utility function over the outcomes
of the lottery. So X= (x1, ..., xK) then bernouilli function is:
u:X→R+by considering all the axioms that hold for utility
functions.
Expected utility function: Is a utility function on the set
of ∆(X) of utilities.
Bernouilli function / von Neumann morgenstern
utility function: If is a binary relation on X representing
the agent’s preferences over lotteries over T. If there is a
function v:T→Rsuch that
xy⇔
m
X
k=1
xkv(τk)≥
m
X
k=1
ykv(τk)
then
u(x) =
m
X
k=1
xkv(τk)
where v is called a Bernouilli function, and where xiare the
probabilities of event τihappening
Existence of Neumann-Morgenstern utility function:
Let be a complete, transitive and continuous preference
