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Non- Cooperative Game Theory, Cooperative game, Preferences and utility, Games, Extensive form, Evolutionary Game Theory, Normal form games and more in this cheat sheet
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Players: N = 1, 2 , .., n Actions/strategies: Each player chooses si from his own finite strategy set: Si for each i ∈ N resulting in a tuple that describes strategy combination: s = (s 1 , ..., 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 ∈ Si such that
ui(s∗ i , si) ≥ ui(s‘ i, s−i)∀s‘ i ∈ Siands‘ i 6 = s∗ i
Pure - strategy (Nash Equilibrium): All strategies are mutual best responses:
ui(s∗ i , si) ≥ ui(s‘ i, s−i)∀s‘ i ∈ Siands‘ i 6 = s∗ i
Population of players: N = 1, 2 , ..., n Colations: C ⊂ N form in the population and become players results in a coalition structure: ρ = {C 1 , C 2 , ..., Ck } Payoffs: Φ = {Φ 1 , ..., Φn} and we need a sharing rule for for the individual player resulting in: Φi = Φ(ρ, ”sharing rule”) characteristic function form (CFG): The Game is defined by 2-tuple G(v, N ) where Characteristic function: v : 2N^ → R where 2N^ are 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:
i∈C Φi^ ≤^ 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 = (Φ 1 , ..., Φn) are:
i∈N Φi^ =^ v(N^ ) so whole value is out
i∈C Φi^ ≥^ 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:
i∈C α(Ci) = 1
C∈ 2 N^ α(C)v(C)
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 allocation/value x∗(v) should satisfy:
i∈N x
∗ i (v) =^ v(N^ )
S∈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 = C 1 , C 2 , ..., Ck implies 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 ∈ M and all wi ∈ W are single. 2 Engage) Each single man m ∈ M proposes to his preferred woman w to 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 proposer m over m’ she becomes engaged with m and m’ becomes single. If not (m’, w) remain engaged. c) All proposers 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
A binary relation (weakly prefers), (prefers), ∼ (indifferent) on a set X is a non-empty subset P ⊂ X × X. We write x y iff (x, y) ∈ P Assumptions on preferences:
A utility function for a binary relation on a set X is a function u : X → R such that
u(x) ≥ u(y) ⇔ x y
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 = (x 1 , ..., xK ) then a lottery is represented by (p 1 , ..., 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 := (p 1 , ..., pk ) and p′^ := (p′ 1 , ..., p′ k ), p p′^ only when U (p) ≥ U (p′) Bernouilli function is the utility function over the outcomes of the lottery. So X = (x 1 , ..., 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 → R such that
x y ⇔
∑^ m
k=
xk v(τk ) ≥
∑^ m
k=
yk v(τk )
then
u(x) =
∑^ m
k=
xk v(τk )
where v is called a Bernouilli function, and where xi are the probabilities of event τi happening Existence of Neumann-Morgenstern utility function: Let be a complete, transitive and continuous preference
relation on X = ∇(T ) for any finite set T. Then admits a utility function u of the expected-utility form iff meets the axiom of independence of irrelevant alternatives. Sure thing principle (Savage): A decision maker who would take a certain Action A if he knew that event B happens should also take Action A if he new that B not happens and also if he knew nothing about B. (This is equivalent to independence of irrelevant alternatives) Risk neutral: An agent is risk-neutral iff he is indifferent between accepting and rejecting all fair gambles that is for all α, τ 1 , τ 2 :
E[u(lottery)] = α · v(τ 1 ) + (1 − α) · v(τ 2 ) = u(ατ 1 + (1 − α)τ 2 )
Risk averse: An agent is risk averse iff he rejects all fair gambles for all α, τ 1 , τ 2 :
E[u(lottery)] = α · v(τ 1 ) + (1 − α) · v(τ 2 ) < u(ατ 1 + (1 − α)τ 2 )
Since g(λα + (1 − λ)β) > λg(α) + (1 − λ)g(β) is the def. of concavity to be risk averse the utility function has to be strictly concave. Risk seeking: An agent is risk seeking iff he strictly prefers all fair gambles for all α, τ 1 , τ 2 :
E[u(lottery)] = α · v(τ 1 ) + (1 − α) · v(τ 2 ) > u(ατ 1 + (1 − α)τ 2 )
Since g(λα + (1 − λ)β) < λg(α) + (1 − λ)g(β) is the def. of convexity to be risk seeking the utility function has to be strictly convex.
Normal form:
Normal form triplet: G = (N, {Si}i∈N , {ui}i∈N ) Strategy profile: s = (s 1 , ..., sn) is called a strategy profile. Is a collection of strategies, one for each player. If s is played, player i receives ui(s) Opponents strategies: Write s−i for all strategies except for the one of player i. So a strategy profile may be written as s = (si, s−i) Dominance:
So obvs. we do not play a dominated strategy no matter what others are doing. Dominant-Strategy Equilibrium: The strategy profile s∗ is a dominant-strategy equilibrium if for every player i, ui(s∗ i , s−i) > ui(si, s−i) for all strategy profiles s = (si, s−i) Nash Equilibrium: is a strategy profile s∗^ such that for every player i,
ui(s∗ i , s∗−i) ≥ ui(si, s∗−i)∀si
So no player has any regrets hi could not have done better when all other played like they have. Best reply function: Bi(s−i) = {si|ui(si, s−i) ≥ ui(si‘, s−i)∀s‘ i given the actions from our opponents chose our best action. and with the best function the Nash equilibrium gets: s∗^ is a Nash equilibrium iff s∗ i ∈ Bi(s∗−i)∀i Mixed strategy: A mixed strategy σ 1 for a player i is any probability distribution over his or her set Si of pure strategies. The set of mixed strategies is:
δ(Si) = {xi ∈ R| +S i|:
h∈Si
xih = 1
Mixed extension The mixed extension of a game G has players, strategies and payoffs: Γ = (N, {Si}i∈N , {Ui}i∈N ) where
s ui(s)^
j∈N σj^ (sj^ ) All the things like best response and nash equilibrium hold also for mixed strategies! How to find mixed Nash equilibria:
(a) Identify candidates: i. if there is such an equilibrium then each of these strategies must yield the same expected payoff given column’s equilibrium strategy. ii. Write down these payofss ans olve for column’s equilibrium mix. iii. Reverse: Look at the strategies that column is mixing on and solve for rows equilibrium mix. (b) Check candidates:
i. The equilibrium mix we found must indeed involve the strategies for row we started with. ii. All probabilities we found must indeed be probabilities (between 0 and 1)
iii. Neither player has a positive deviation
Nash Theorem: Every finite game has at least one ”Nash” Equilibrium in mixed strategies!
Trembling hand / perfect equilibrium: Take the strategy that is played by nash equilibrium. E.g. (a1, b1) then check if E[u(a1)] = u(a 1 , b1)(1 − ) + u(a 1 , b2) where b2 is played with some error probability . And despite the error this must be greater then as if player 1 would play a2 instead: E[u(a2)] = u(a 2 , b1)(1 − ) + u(a 2 , b2) if E[u(a1)] > E[u(a2)] the nash equilb. is perfect equilibrium. Extensive form game: Players, Basic structure is a game tree with nodes a ∈ A, a 0 is root of tree. And nodes can bee Decision nodes where a player makes a decision or a Chance node where nature plays according to some probability distribution. (If 2 nodes are connected we have no information from above and we have to decide based on nash equilibrium. ) Subgames if a node has been reached we have full information at its root node and we can decide as if we’re in isolation. Strategy set in subgame: A strategy for a player is over the whole game so if we have 2 subgames with possibility a,b then we have (a,a), (a, b), (b, a), (b, b) where we simultaneously describe as we would play both subgames despite the fact that we play only 1. (important if we have the same information in 2 subgames (shown via connection in between the subnodes) then we have only 2 strategies(a,b).
Symmetric two-player games: G = (N, {Si}i∈N , {ui}i∈N.
Symmetric Nash Equilibrium: is a strategy profile σ∗^ such that for every player i,
ui(σ∗, σ∗) ≥ ui(σ, σ∗)∀σ
, ”if no player has an incentive to deviate from their part in a particular stategy profile, then it is Nash Equilibrium. In a symmetric normal form game there always exists a symmetric Nash Equilibrium. (Not all Nash Equilibria of a symmetric game need to be symmetric. Evolutionary stable strategy(ESS): A mixed strategy σ ∈ δ(S) is an evolutionary stable strategy (ESS) if for every strategy τ 6 = σ there exists (τ ) ∈ (0, 1) s.t. ∀ ∈ (0, (τ )) :
U (σ, τ + (1 − )σ) > U (τ, τ + (1 − )σ)