wass course on game theory - pvmouche.deds.nl · wass course on game theory pierre von mouche...

Introduction Recalling games in strategic form and convexity Semi-uniqueness and marginal reductions Coalition formation

WASS Course on Game Theory

Pierre von Mouche

December 13 – 14, 2011Wageningen

Organisation

Motivating aggregative games (slides).Recallling relevant standard theory of games in strategicform (onb blackboard, hard copy provided).Recalling relevant convexity notions (on blackboard).Semi-uniqueness via marginal reductions (slides).Interesting application: coalitional formation (slides).

In between: exercises (hard copies provided).

Organisation

Motivating aggregative games (slides).Recallling relevant standard theory of games in strategicform (onb blackboard, hard copy provided).Recalling relevant convexity notions (on blackboard).Semi-uniqueness via marginal reductions (slides).Interesting application: coalitional formation (slides).

In between: exercises (hard copies provided).

What is an aggregative game?

An aggregative game is a game in strategic form between nplayers where each player i ∈ {1, . . . ,n} has as strategy set asubset X i of R and the payoff function f i has the form

f i(x1, . . . , xn) = πi(x i ,

n∑l=1

where, with Y :=∑n

l=1 X l (Minkowski-sum), πi : X i × Y → R.πi is called reduced payoff function of player i .

What is an aggregative game? (ctd.)

More general:Weights: f i(x1, . . . , xn) = πi(x i ,

∑nl=1 tlx l) with the tl ≥ 0.

Co-strategy mappings:f i(x1, . . . , xn) = πi(x i , ϕi(x1, . . . , xn))

variants with higher dimensional strategy sets.Most important case: each X i is an interval of R.

Many economic games are aggregative

Heterogeneous cournot oligopoly

πi(q1, . . . ,qn) = pi(q1 + · · ·+ qn)qi − c i(qi).

Homogeneous if p1 = · · · = pn.Heterogenous bertrand duopoly

πi(p1, . . . ,pn) = pi f i(p1, . . . ,pn)− c i(f i(p1, . . . ,pn)).

Formal transboundary pollution game

f i(x1, . . . , xn) = P i(x i)−Di(Ti1x1 + · · ·+ Tinxn).

Many economic games are aggregative (ctd.)

Pure public good game

f i(x1, . . . , xn) = ui(mi − x i , x1 + · · ·+ xn).

Joint production game, search game, ... (seeprepublication of Cornes and Hartley).

Many economic games are aggregative (ctd.)

Pure public good game

f i(x1, . . . , xn) = ui(mi − x i , x1 + · · ·+ xn).

Joint production game, search game, ... (seeprepublication of Cornes and Hartley).

’Grand project’

’Grand project’: understanding the geometric structure of theset of Nash equilibria E of aggregative games.

Three steps working plan:1 unifying results on (geometric structure of) Cournot

equilibria; Cournot oligopolies are not so realistic games,but provide a ’paradise’ on techniques; ALMOST DONE

2 intrinsically improving the unified results (using generalisedconvexity results); ALMOST DONE

3 generalisation to aggregative games. IN PROGESSFuture: understanding the pareto efficiency/inefficiency ofequilibria.

’Grand project’

Basic questions

In order to understand the geometric structure in particular thefollowing questions arise:

Existence: does there exists a nash equilibrium?Semi-uniqueness: does there exists at most one nashequilibrium?Uniqueness: does there exists a unique nash equilibrium?

Note on existence

Almost all avaible uniqueness results for aggregativegames can be proved by a variant of the Nikaido-Isodatheorem (see notes).In these results conditional payoff functions arequasi-concave.However, there are (deep) existence results based onother principles (without uniqueness).So in uniqueness results only semi-uniqueness is an issue.

General semi-uniqueness results for games in strategic form donot really apply to aggregative games.

Note on existence

On blackboard; see hardcopy.

Two techniques

Techniques for aggregative games:Non-linear and convex programming.Backward best reply correspondance.

For existence and uniqueness: whole correspondance.For semi-uniqueness: special property of marginalreductions. Discovered by Corchón.

Idea illustrated for transboundary pollution game

The following transboundary pollution game (global case) hasat most one interior equilibrium:

X i = [0,mi ];f i(x) = P i(x i)−Di(x1 + · · ·+ xn);all Di , P i differentiable;all Di convex, P i strictly concave.

Illustration of Proof of Corchón’s Result

1 By contradiction: let a,b two different interior Nashequilibria.

2 Let ya :=∑N

l=1 al , yb :=∑N

l=1 bl . Suppose w.r.g. ya ≥ yb.3 P i ′(ai) = Di ′(ya) and Di ′(yb) = P i ′(bi).4 Di ′ increasing: Di ′(ya) ≥ Di ′(yb).5 P i ′(ai) ≥ P i ′(bi).6 P i ′ is strictly decreasing: ai ≤ bi .7 ya ≤ yb.8 ya = yb.9 P i ′(ai) = P i ′(bi).

10 P i ′ is strictly decreasing, so ai = bi .11 a = b, a contradiction.

2 Let ya :=∑N

l=1 al , yb :=∑N

2 Let ya :=∑N

l=1 al , yb :=∑N

2 Let ya :=∑N

l=1 al , yb :=∑N

2 Let ya :=∑N

l=1 al , yb :=∑N

2 Let ya :=∑N

l=1 al , yb :=∑N

2 Let ya :=∑N

l=1 al , yb :=∑N

2 Let ya :=∑N

l=1 al , yb :=∑N

2 Let ya :=∑N

l=1 al , yb :=∑N

2 Let ya :=∑N

l=1 al , yb :=∑N

2 Let ya :=∑N

l=1 al , yb :=∑N

More general result

Theorem

Consider a game in strategi form (X 1, . . . ,X n; f 1, . . . , f n).Sufficient for existence of at most one Nash equilibrium is thatfor each player i

X i is a non-degenerate interval of R;Di f i(x1, . . . , xn) = t i(x i , x1 + · · ·+ xn);the marginal reduction t :X i × Y → R

is strictly decreasing in x i ;and decreasing in y.

Suppose a,b ∈ E .Step 1. We have

a ≥ b ⇒ a ≤ b.

Proof: by contradiction. So suppose j with aj > bj . As a ∈ Eand aj is not a left boundary point of X j , Dj f j(a) ≥ 0.In thesame way Dj f j(b) ≤ 0. Noting that

Dj f j(a) = t j(aj ,a), Dj f j(b) = t j(bj ,b),

we havet j(aj ,a) ≥ 0 ≥ t j(bj ,b).

The monotonicity properties of t j imply the contradiction

t j(aj ,a) ≤ t j(aj ,b) < t j(bj ,b).

a ≥ b ⇒ a ≤ b.

Proof (ctd.)

Step 2. We havea ≥ b ⇒ a ≤ b.

Proof: suppose a ≥ b. By Step 1, a ≤ b. Thus a ≤ b.

Step 3. We havea = b.

Proof: a ≥ b or b ≥ a. We may assume a ≥ b. By Step 1,ai ≤ bi (i ∈ N). So a ≤ b. Thus a = b.

Proof (ctd.)

Step 2. We havea ≥ b ⇒ a ≤ b.

Proof: suppose a ≥ b. By Step 1, a ≤ b. Thus a ≤ b.

Step 3. We havea = b.

Proof: a ≥ b or b ≥ a. We may assume a ≥ b. By Step 1,ai ≤ bi (i ∈ N). So a ≤ b. Thus a = b.

Generalisations

Two generalisations possible:For non-differentiable case.For coalitional equilibria.

Coalition formation

So-called ‘new approach’ of coalition formation. Thisapproach consists in modeling coalition formation as atwo-stage game; the goal is to determine equilibriumcoalition structures.Roughly speaking, in the first stage, each player i choosesa membership action which leads to a coalition structure.In the second stage each coalition in this coalition structurechooses its ‘physical’ actions.

Coalition formation (ctd.)

Suppose N = {1, . . . ,n} is a set players.Let C? be the set of possible coalition structures with theseplayers.If Γ is a game in strategic form and C ∈ C?, then Γ(C)denotes the game in strategic form with

as players, also referred to as ’meta-players’, the coalitionsin C,with for each meta-player as strategy set the cartesianproduct of the strategy sets in Γ of the members of themeta-player,with the payoff function of a meta-player the aggregatepayoff of its members.

The Nash equilibria of Γ(C) are called coalitional equilibria.

With the above notations we now can formulate the two-stagegame. Formally, the fundamental objects are

a set of players N,a game in strategic form Gb for these players,for each player i a set Mi and, with M :=

∏ni=1 M i ;

a so-called membership rule R : M→ C?;(and may be a sharing rule S).

∏ni=1 M i ;

The game Gb is referred to as the base game; in this gamethe physical actions are taken. It is assumed that eachgame Gb(C) has a unique coalitional equilibrium; let V i(C)be the individual payoff of player i in this equilibrium (maybe after applying the sharing rule). A weaker appropriateassumption would be: each Gb(C) has at least onecoalitional equilibrium and for each player it holds that hehas the same payoff in each of these equilibria.V := (V 1, . . . ,V n) : C? → Rn is called valuation and thusassigns to each coalition structure C an individual payoffV i(C) for each player i .

Out of these fundamental objects the game in strategic form G?

is defined as follows:Player set is N;The strategy set for player i is M i and his payoff function isf i := V i ◦ R;The Nash equilibria of G? induce via the membership ruleR the equilibrium coalition structures.

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