Advanced Microeconomics P . v. Mouche Wageningen University 2016 - - PowerPoint PPT Presentation

Microeconomics Game Theory

Advanced Microeconomics

P . v. Mouche

Wageningen University

2016

Microeconomics Game Theory

Outline

1

Microeconomics Motivation Reminder Core Walrasian equilibrium Welfare theorems Quiz

2

Game Theory Motivation Games in strategic form Games in extensive form

Microeconomics Game Theory

Capitalism

According to Adam Smith’s (the Wealth of nations 1776): a laissez-faire approach to economics is the essential way to ensure prosperity for a nation as a whole. Ultimately, when capitalism is allowed to run its course, the greed and self-interest of the capitalists would produce results in the economy that benefit not only the individual, but society as well. Scientific problem: prove such claims with models. This problem comes down to: general equilibrium exists, is efficient and stable.

Microeconomics Game Theory

Overall notations

Space of goods Rn

+ = {x = (x1, . . . , xn) ∈ Rn | x1 ≥ 0, . . . , xn ≥ 0}.

Utility function u : Rn

+ → R.

Prices p = (p1, . . . , pn). All pi > 0 (if not stated otherwise). Note: prices do not appear in the utility function. Budget (income) m ≥ 0. (Marshallian) demand correspondences ˜ xi(p; m). Often these are functions. Producer theory will be less important for what follows.

Microeconomics Game Theory

Concrete functions

Cobb-douglas: u(x) = Axα1

1 · · · xαn n ,

˜ xi(p; m) = αi α1 + · · · + αn m pi . Ces: u(x) = A(α1xρ

1 + · · · + αnxρ n)h/ρ.

Microeconomics Game Theory

Concrete functions (cont.)

Leontief: u(x) = min (x1/α1, . . . , xn/αn). Example What are the marshallian demand functions for this utility function? Answer: ˜ xi(p; m) = αim α1p1 + · · · + αnpn .

Microeconomics Game Theory

Concrete functions (cont.)

Solow (for n = 2): u(x1, x2) = α1x1 + α2x2. Here there are marshallian demand correspondences (instead

• f functions).

Maximum (for n = 2): u(x1, x2) = max (α1x1, α2x2).

Microeconomics Game Theory

Increasingness

Remember the notations ≤, <, ≪. For instance: (3, 4) ≤ (3, 4), (3, 4) ≤ (3, 5). (3, 4) < (3, 5), (3, 4) < (4, 5). (3, 4) ≪ (4, 5). Remember: u is increasing: x ≤ y ⇒ u(x) ≤ u(y). u is strongly increasing: x < y ⇒ u(x) < u(y). u is strictly increasing: u is increasing, and x ≪ y ⇒ u(x) < u(y). Example Is the cobb-douglass utility function strongly increasing? Answer: no (but it is strictly increasing).

Microeconomics Game Theory

Setting for core

Pure exchange economy. Specified by: N consumers and n goods. for each consumer h a good bundle ωh = (ωh

1, . . . , ωh n) > 0

(i.e. each consumer has something), to be called initial good bundle such that for each good k, Ok :=

N

• h=1

ωh

k > 0

(i.e. each good is present). for each consumer h a continuous utility function uh : Rn

+ → R.

Attention: there are (still) no prices!

Microeconomics Game Theory

Allocations

Allocation: X := (x1, . . . , xN) ∈ (Rn

+)N.

Initial allocation: Ω := (ω1, . . . , ωN). Feasible allocation: N

h=1 xh k = Ok (1 ≤ k ≤ n). Attention:

not ≤-sign! Feasible allocation is interior if 0 < xh

k < Ok for all h and k.

Microeconomics Game Theory

Pareto efficiency

Definition A feasible allocation X is called weakly pareto efficient if there is no feasible allocation Y with uh(yh) > uh(xh) (1 ≤ h ≤ N). (strongly) pareto efficient if there is no feasible allocation Y with uh(yh) ≥ uh(xh) (1 ≤ h ≤ N) with at least one of these inequalities strict.

Microeconomics Game Theory

Pareto efficient allocations

Each strongly pareto efficient allocation is also weakly pareto efficient. So each weakly pareto inefficient allocation is strongly pareto inefficient. In fact weakly and strongly pareto efficiency make sense in

• ther contexts. (See next Example.)

Microeconomics Game Theory

Example

Example Consider N agents for which there are a finite number of ’states

• f the world’. Denote by (a1, a2, . . . , aN) a state where agent i

has ‘utility’ ai. Determine for the following situations which states are strongly pareto efficient and which are weakly pareto efficient.

• a. A = (5, 10), B = (6, 9), C = (6, 11), D = (4, 12).

Answer: Weak: B, C, D. Strong: C, D.

• b. A = (6, 6), B = (6, 7), C = (3, 2), D = (7, 6), E =

(5, 6), F = (11, 1). Answer: Weak: A, B, D, F. Strong: B, D, F.

• c. A = (5, 4), B = (9, 1), C = (3, 8).

Answer: Weak: A, B, C. Strong: A, B, C.

Microeconomics Game Theory

Example (cont.)

Example

• d. A = (−4, 8), B = (−4, 3), C = (−5, −3), D = (6, 0).

Answer: Weak: A, B, D. Strong: A, D.

• e. A = (1, 2, 6, 4), B = (4, 8, 3, 2), C = (1, 8, 1, 2), D =

(0, 0, 0, 0). Answer: Weak: A, B, C. Strong: A, B.

• f. A = (1, 3, 5), B = (1, 3, 5), C = (2, 4, 3).

Answer: Weak: A, B, C. Strong: A, B, C.

• g. A = (1, 3, 5), B = (1, 3, 5), C = (2, 4, 3), D = (1, 3, 6).

Answer: Weak: A, B, C, D. Strong: C, D.

• h. A = (1), B = (−8), C = (137).

Answer: Weak: C. Strong: C.

Microeconomics Game Theory

Strong versus weak pareto efficiency

Theorem If each utility function is continuous and strongly increasing, then the set of weak and strong pareto efficient allocations is the same. Proof. This is a technical result. We omit here its proof. (If wished, see exercise 5.44 in the text book).

Microeconomics Game Theory

Pareto

Microeconomics Game Theory

Pareto (ctd.)

Vilfredo Pareto (1848-1923): Italian engineer, economist and sociologist. Very good knowledge of mathematics. For 20 years director of two Italian railway companies. Later, motivated by Walras to switch to economic research. After disenchantment in economics, switched to sociology. His articles are difficult to read.

Microeconomics Game Theory

Barter equilibrium

What would be a reasonable feasible allocation X = (x1, . . . , xN) when the N consumers exchange goods?

• 1. X is individually rational.

This is defined as follows: X is individually rational for consumer h, if uh(xh) ≥ uh(ωh) and X is individually rational if X is for each consumer individually rational. Is that all? If an allocation is strongly pareto inefficient, then there is another feasible allocation making someone better off and no

• ne worse off: then a trade can be arranged to which no

consumer will object. So: 2 X is strongly pareto efficient. Is that all?

Microeconomics Game Theory

Core

NO: 3 X should belong to the core.

Microeconomics Game Theory

Definition Let X = (x1, . . . , xN) be a feasible allocation. We say that a coalition S can improve upon x if there are good bundles yh (h ∈ S) such that

1

• h∈S yh =

h∈S ωh;

2

uh(yh) ≥ uh(xh) for all h ∈ S with at least one inequality strict. Definition A feasible allocation belongs to the (strong) core if there is no coalition that can improve upon it. Attention: the set of pareto efficient allocations only depends on O1, . . . , On, but the core depends on the whole initial allocation!

Microeconomics Game Theory

Properties of core

Theorem Each element of the core is individually rational and pareto efficient. Proof. Suppose X is in the core. S = {i} cannot improve upon X. In particular for yi = ωi it does not hold that uh(yh) ≥ uh(xh) for all h ∈ S with at least one inequality strict. So ui(yi) ≤ ui(xi), i.e. ui(ωi) ≤ ui(xi) follows. Thus X is individual rational for consumer player i. As this holds for every i, X is individual rational. Also S = {1, . . . , N} cannot improve upon X. So for any feasible allocation Y it does not hold that uh(yh) ≥ uh(xh) for all h ∈ S with at least one inequality strict, i.e. that Y is a pareto improvement of X. Thus X is pareto-efficient.

Microeconomics Game Theory

Core for N = 2

Theorem For N = 2 the core is the set of pareto efficient individually rational allocations. Proof. Because S = {1}, S = {2} or S = {1, 2}.

Microeconomics Game Theory

Non-empty core?

Very fundamental question: is the core non-empty? Intuitively one may think that this is always the case. However, this is not true. (Not so easy to find good counter-examples.) We shall see: Theorem Each pure exchange economy where each utility function is continuous, strictly quasi-concave and strongly increasing has a non-empty core. Remark: result also holds for cd-function (see Exercise 5.14 in the text book).

Microeconomics Game Theory

Box of Edgeworth

Box of Edgeworth: D := {x ∈ R2 | 0 ≤ xk ≤ Ok (1 ≤ k ≤ 2)}. One can identify a feasible allocation with the corresponding point in D. The set of pareto efficient allocations in the box is the contract curve.

Microeconomics Game Theory

Pareto efficient allocations: necessary condition

Theorem If X = (x1, . . . , xN) is an interior pareto efficient allocation, then under mild differentiability conditions, equality for each consumer of each specific marginal rate of substitution holds. Proof. We omit the proof which can be given with the method of Lagrange and only illustrate with a figure the idea.

Microeconomics Game Theory

Pareto efficient allocations: sufficient condition

Theorem If each utility function is quasi-concave, then under mild differentiability conditions each interior feasible allocation where for each consumer each specific marginal rate of substitution is the same, is pareto efficient. Proof. We omit the proof and only illustrate with a figure the idea.

Microeconomics Game Theory

Example

Example Determine the contract curve for: uA = xα1

1 xα2 2 , uB = xγ1 1 xγ2 2 ,

where α1 + α2 = γ1 + γ2 = 1. Answer: xA

2 =

α2γ1O2xA

1

α1γ2O1 + (γ1 − α1)xA

1

, where 0 ≤ xA

1 ≤ O1. (In fact the points on the boundary of the

box need special investigation.)

Microeconomics Game Theory

Walrasian equilibrium

Three mechanisms for price formation: authority, trade and

• auctions. We further only here deal with trade.

So allow for markets and (may be zero) prices: p ∈ Rn

+

Definition (p; X) is a Walrasian equilibrium if – for each h, xh is a maximiser of the utility function uh : Rn

+ → R under the restriction p · x ≤ p · ωh;

– X is feasible; i.e. N

h=1 xh k = Ok (1 ≤ k ≤ n).

p: equilibrium price vector. X: equilibrium allocation.

Microeconomics Game Theory

Equilibrium and aggregate excess demand

Suppose that each consumer has well-defined marshallian demand functions. For this situation we can define the notion of aggregate excess demands z1, . . . , zn as follows: zi(p) :=

N

• h=1

eh

i (p) aggregate excess demand of good i,

eh

i (p) := ˜

xh

i (p; mh)−ωh i excess demand of good i for consumer h ,

where mh := p · ωh. Now: p ∈ Rn

++ is an equilibrium price vector if and only if for the

aggregate excess demands zk of the goods one has z1(p) = · · · = zn(p) = 0.

Microeconomics Game Theory

Fundamental observations

Each equilibrium allocation is individually rational. If p is an equilibrium price vector, then for each λ > 0 also λp is. Law of Walras: p1z1(p) + · · · + pnzn(p) = 0 for every p. Law of Walras implies that if n − 1 aggregate excess demands are zero, then all are zero.

Microeconomics Game Theory

Existence of equilibrium

Theorem Each pure exchange economy where each utility function is continuous, strictly quasi-concave and strongly increasing has a walrasian equilibrium with positive equilibrium prices. Proof. Very complicated. (See text book.)

Microeconomics Game Theory

Notes on the proof

Strict quasi-concavity and continuity guarantee that there are well-defined marshallian demand functions. Proving that there exists an equilibrium price vector can be done with Brouwers’ fixed point theorem. In fact Debreu presented a complete proof. The proof also allows for prices zero. A quite technical problem in the proof is to be sure that prices are not 0.

Microeconomics Game Theory

Proof by not correct principle

Principle: n equations in n variables have a solution. Applied to our equilibrium existence problem: zi(p) = 0 (1 ≤ i ≤ n) are n equations in n variables. However: according to Law of Walras we only have n − 1 equations, so principle does not apply. But according to p equilibrium ⇒ λp equilibrium we have n − 1 equations. So principle applies.

Microeconomics Game Theory

Brouwer and his fixed point theorem

Microeconomics Game Theory

Brouwer and his fixed point theorem (cont.)

Luitzen Jan Egbertus Brouwer (1881-1966). Dutch mathematician, Frisian and idealist. Brouwer proved a number of theorems that were breakthroughs in the emerging field of topology. Most famous is his fixed point theorem. He died after he was strucked by a vehicle while crossing the street in front of his house. Fixed point theorem of Brouwer: each continuous function f from the unit ball in Rn to itself has a fixed point, i.e. there is x such that f(x) = x. For one dimension the theorem is not so deep.

Microeconomics Game Theory

Debreu

Gérard Debreu (1921-2004): French economist and mathematician. Wrote ’Theory of Value’ for his PhD thesis. Nobel price for economics in 1983 together with Arrow.

Microeconomics Game Theory

Example

Example Determine an equilibrium price vector for uA(x1, x2) = x1x2, uB(x1, x2) = min (x1, x2), ωA = (5, 0), ωB = (0, 6). Answer: p2/p1 = 5/7.

Microeconomics Game Theory

First welfare Theorem

Theorem (First welfare Theorem.) Consider a pure exchange economy where each utility function is continuous and locally non-satiated. Then: each equilibrium allocation is pareto efficient. Proof. Proof, with additional assumptions, possible with Gossen’s second law.

Microeconomics Game Theory

Perfect proof of theorem

By contradiction: suppose X is an equilibrium allocation that is pareto inefficient. Let (p; X) be a walrasian equilibrium. Because X is pareto inefficient, there exists a feasible allocation Y and a consumer, say j, such that uh(yh) ≥ uh(xh) for all h uj(yj) > uj(xj). Because xj is a maximiser of uj under the restriction p · x ≤ p · ωj, and by the above yj is not, it follows that p · yj > p · ωj.

Microeconomics Game Theory

Perfect proof of theorem (cont.)

Also for each h p · yh ≥ p · ωh. (∗) Indeed: because if not, then there is a k with p · yk < p · ωk. Because uk is in yk locally non-satiated, there exists a with p · a < p · ωk and uk(a) > uk(yk). But then p · a < p · ωk and uk(a) > uk(xk), a contradiction. Adding now the N inequalities (*) with strict inequality for h = j we obtain: p ·

N

• h=1

yh > p ·

N

• h=1

ωh, a contradiction with feasibility of Y.

Microeconomics Game Theory

Improvement of first welfare theorem

Theorem Consider a pure exchange economy where each utility function is continuous and locally non-satiated. Then: each equilibrium allocation is in the core. Proof. Adapt proof of First welfare Theorem. (See text book for the details) Thus (in the case of continuous locally non-satiated utility functions) the core is not empty if walrasian equilibrium exists.

Microeconomics Game Theory

Second welfare Theorem

Theorem (Second welfare theorem.) Consider a pure exchange economy where each utility function is continuous, strongly increasing and strictly quasi-concave. Let X be a pareto efficient

• allocation. Then the pure exchange economy where X is the

initial allocation has X as unique equilibrium allocation.

Microeconomics Game Theory

Perfect proof

Proof. Consider the pure exchange economy where X is the initial allocation. Also here X is pareto efficient. We know: there exists a walrasian equilibrium (p; Y). This implies, for all h, uh(yh) ≥ uh(xh). Because X is pareto efficient, even uh(yh) = uh(xh) for all h. We show by contradiction Y = X.

Microeconomics Game Theory

Perfect proof

Proof. So suppose yr = xr for some r. Then, as ur is strictly quasi-concave ur(1 2(xr + yr)) > min (ur(xr), ur(yr)) = ur(yr). Because p · yr ≤ p · xr, we have p · 1 2(xr + yr) = 1 2(p · xr) + 1 2(p · yr) ≤ p · xr, a contradiction.

Microeconomics Game Theory

Summary

First results on general equilibrium obtained by Walras. Debreu made these results mathematical rigorous using Brouwer’s fixed point theorem. (Under weak conditions) each equilibrium is in the core. Not each element of the core has to be an equilibrium. As the economy becomes larger and larger, the core shrinks to include only equilibrium allocations. Method of proof by concept of ’replica economy’. (Section 5.5 in text book.) Proof first provided by Debreu and Scarf.

Microeconomics Game Theory

Little quiz

Consider a pure-exchange economy.

• 1. The assumption Ok > 0 means that every consumer has a

positive amount of good type k. False

• 2. There is at least one producer. False
• 3. The consumers have market power. False
• 4. Each weakly pareto inefficient allocation is strongly pareto
• inefficient. True
• 5. The initial allocation is pareto inefficient. False

Microeconomics Game Theory

Little quiz (ctd.)

• 6. If each utility function is continuous and strongly

increasing, then there exists a walrasian equilibrium. False

• 7. It is possible that there exists a walrasian equilibrium

where each consumer has utility 0. True

• 8. If each utility function is continuous and strongly

increasing, then the set of weak and strong pareto efficient allocations is the same. True

• 9. The core depends on the initial allocation. True
• 10. Every pareto efficient individual rational allocation belongs

to the core. False

Microeconomics Game Theory

Little quiz (ctd.)

• 11. If there exists a walrasian equilibrium, then there exist

more then two equilibrium price vectors. True Now suppose N = n = 2 and uA(x1, x2) = x1x2, uB(x1, x2) = x1/3

1

x2/3

2

, ωA

1 = 2, ωA 2 = 2, ωB 1 = 3, ωB 2 = 1.

• 12. The allocation X =
• (1, 2); (1, 4)
• is pareto efficient. False
• 13. The allocation ((5, 0), (0, 3)) is pareto efficient. False

Microeconomics Game Theory

What is game theory?

Traditional game theory deals with mathematical models of conflict and cooperation in the real world between at least two rational intelligent players. Player: humans, organisations, nations, animals, computers, Situations with one player are studied by the classical

• ptimisation theory.

’Traditional’ because of rationality assumption. ’Rationality’ and ’intelligence’ are completely different concepts.

Microeconomics Game Theory

Nature

Applications: parlor games, military strategy, computer games, biology, economics, sociology, psychology antropology, politicology. Game theory provides a language that is very appropriate for conceptual thinking. Many game theoretical concepts can be understood without advanced mathematics.

Microeconomics Game Theory

Outcomes and payoffs

A game can have different outcomes. Each outcome has its own payoffs for every player. Nature of payoff: money, honour, activity, nothing at all, utility, real number, ... . Interpretation of payoff: ‘satisfaction’ at end of game. In general it does not make sense to speak of winners and losers.

Microeconomics Game Theory

Rationality

Because there is more than one player, especially rationality becomes a problematic notion. For example, what would You as player 1 play in the following bi-matrix-game: 300; 400 600; 250 200; 600 450; 500

• .

(One player chooses a row, the other a column; first (second) number is payoff to row (column) player.)

Microeconomics Game Theory

Tic-tac-toe

Notations: 1 2 3 4 5 6 7 8 9 Player 1: X. Player 2: O. Many outcomes (more than three). Can, for player 1, be

• rdered by player 1 wins, draw, player 1 looses. It is a

zero-sum game. Payoffs (example): winner obtains 13 Euro from looser. When draw, then each player cleans the shoes of the other. Example of a play of this game:

Microeconomics Game Theory

Tic-tac-toe (cont.)

X X O X X O X X O O X X O X O X X O O X O So: player 2 is the winner. Question: Is player 1 intelligent? Is player 1 rational? Answer: We do not know.

Microeconomics Game Theory

Real-world types

all players are rational – players may be not rational all players are intelligent – players who may be not intelligent binding agreements – no binding agreements chance moves – no chance moves communication – no communication static game – dynamic game transferable payoffs – no transferable payoffs interconnected games – isolated games (In red what we will assume always.) perfect information – imperfect information complete information – incomplete information

Microeconomics Game Theory

Perfect information

A player has perfect information if he knows at each moment when it is his turn to move how the game was played untill that moment. A player has imperfect information if he does not have perfect information. A game is with (im)perfect information if (not) all players have perfect information. Chance moves are compatible with perfect information. Examples of games with perfect information: tic-tac-toe, chess, ... Examples of games with imperfect information: poker, monopoly (because of the cards, not because of the die).

Microeconomics Game Theory

Complete information

A player has complete information if he knows all payoff functions. A player has incomplete information if he does not have complete information. A game is with (in)complete information if (not) all players have complete information. Examples of games with complete information: tic-tac-toe, chess, poker, monopoly, ... Examples of games with incomplete information: auctions,

• ligopoly models where firms only know the own cost

functions, ...

Microeconomics Game Theory

Common knowledge

Something is common knowledge if everybody knows it and in addition that everybody knows that everybody knows it and in addition that everybody knows that everybody knows that everybody knows it and ...

Microeconomics Game Theory

Common knowledge

A group of dwarfs with red and green caps are sitting in a circle around their king who has a bell. In this group it is common knowledge that every body is intelligent. They do not communicate with each other and each dwarf can only see the color of the caps of the others, but does not know the color of the own cap. The king says: ”Here is at least one dwarf with a red cap.”. Next he says: “I will ring the bell several times. Those who know their cap color should stand up when i ring the bell.”. Then the king does what he announced. The spectacular thing is that there is a moment where a dwarf stands up. Even, when there are M dwarfs with red caps that all these dwarfs simultaneously stand up when the king rings the bell for the M-th time.

Microeconomics Game Theory

Mathematical types

Game in strategic form. Game in extensive form. Game in characteristic function form.

Microeconomics Game Theory

Game in strategic form

Definition Game in strategic form, specified by n players: 1, . . . , n. for each player i a strategy set (or action set) Xi. for each player i payoff function fi : X1 × · · · × Xn → R. X := X1 × · · · × Xn: set of multi-strategies (or strategy profiles). Interpretation: players choose simultaneously a strategy. A game in strategic form is called finite if each strategy set Xi is finite.

Microeconomics Game Theory

Some concrete games in strategic form

Cournot-duopoly: n = 2, Xi = [0, mi] or Xi = R+ fi(x1, x2) = p(x1 + x2)xi − ci(xi). Transboundary pollution game: n arbitrary, Xi = [0, mi] fi(x1, . . . , xn) = Pi(xi) − Di(Ti1x1 + · · · + Tinxn).

Microeconomics Game Theory

Some concrete games in strategic form (ctd.)

The Hotelling bi-matrix game depends on two parameters: integer n ≥ 1 and w ∈ ]0, 1]. Consider the n + 1 points of H := {0, 1, . . . , n} on the real line, to be referred to as vertices. Two players simultaneously and independently choose a

• vertex. If player 1 (2) chooses vertex x1 ( x2), then:

Case w = 1: the payoff fi(x1, x2) of player i is the number

• f vertices that is the closest to his choice xi; however, a

shared vertex, i.e. one that has equal distance to both players, contributes only 1/2. General case: 0 < w ≤ 1: exactly the same vertices as in the above for w = 1 contribute. Take such a vertex. If it is at distance d to xi, then it contributes wd if it is not a shared vertex, and otherwise it contributes wd/2.

Microeconomics Game Theory

Some concrete games in strategic form (ctd.)

Example n = 7 and w = 1. Action profile ( 5,2 ) : Payoffs: 1 + 1 + 1 + 1 = 4 1 + 1 + 1 + 1 = 4 Action profile ( 0,3 ) : Payoffs 1 + 1 = 2 1 + 1 + 1 + 1 + 1 + 1 = 6

Microeconomics Game Theory

Some concrete games in strategic form (ctd.)

Example n = 7 and w = 1. Action profile ( 2,6 ) : Payoffs: 1 + 1 + 1 + 1 + 1

2 = 4 1 2 1 2 + 1 + 1 + 1 = 3 1 2

Action profile ( 3,3 ) : Payoffs:

1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 = 4 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 = 4

Microeconomics Game Theory

Some concrete games in strategic form (ctd.)

Example n = 5 and w = 1/4: Action profile ( 1,3 ) : Payoffs:

1 4 + 1 + 1 8 = 1 3 8 1 8 + 1 + 1 4 + 1 16 = 1 7 16

Action profile ( 1,4 ) : Payoffs:

1 4 + 1 + 1 4 = 1 1 2 1 4 + 1 + 1 4 = 1 1 2

Microeconomics Game Theory

Normalisation

Many games can be represented in a natural way by normalisation as a game in strategic form. For example, chess and tic-tac-toe: n = 2, Xi is set of completely elaborated plans of playing of i, fi(x1, x2) ∈ {−1, 0, 1}. Questions:

1

Give for each player of tic-tac-toe a completely elaborated plan of playing.

2

How the game will be played?

3

Give an optimal strategy for player 1.

Microeconomics Game Theory

Some concrete games.

⎛ ⎝ 0; 0 −1; 1 1; −1 1; −1 0; 0 −1; 1 −1; 1 1; −1 0; 0 ⎞ ⎠ Stone-paper-scissors

Microeconomics Game Theory

Fundamental notions

Best reply correspondence Ri of player i: X1 × · · · × Xi−1 × Xi+1 × · · · × Xn ⊸ Xi. (Strictly) dominant strategy of a player i: (the) best strategy

• f player i independently of strategies of the other players.

Strongly (or strictly) dominated strategy of a player: a strategy of a player for which there exists another strategy that independently of the strategies of the other players always gives a higher payoff.

Microeconomics Game Theory

Fundamental notions (cont.)

Procedure of iterative (simultaneous) elimination of strongly dominated strategies. Multi-strategy that survives this procedure. If there is a unique multi-strategy that survives the above procedure this multi-strategy is called the iteratively not strongly dominated equilibrium. Nash equilibrium: multi-strategy such that no player wants to deviate from it. Strictly dominant equilibrium: multi-strategy where each player has a strictly dominant strategy.

Microeconomics Game Theory

Nash equilibria

A multi-strategy e = (e1, . . . , en) is a nash equilibrium if and

• nly if for each player i one has

ei ∈ Ri(e1, . . . , ei−1, ei+1, . . . , en). Sometimes can be determined by ∂fi ∂xi = 0 (i = 1, . . . , n)

Microeconomics Game Theory

Example

Determine (if any), the strictly dominant equilibrium, the iteratively not strongly dominated equilibrium (if any) and the Nash equilibria of the game Example ⎛ ⎜ ⎜ ⎝ 2; 4 1; 4 4; 3 3; 0 1; 1 1; 2 5; 2 6; 1 1; 2 0; 5 3; 4 7; 3 0; 6 0; 4 3; 4 1; 5 ⎞ ⎟ ⎟ ⎠ . Answer: no strictly dominant equilibrium. The procedure gives 2; 4 1; 4 4; 3 1; 1 1; 2 5; 2

• . Thus the game does not have an

iteratively not strongly dominated equilibrium. Nash equilibria: (1, 1), (1, 2), (2, 2) and (2, 3) (i.e. row 2 and column

Microeconomics Game Theory

Example

Determine (if any), the strictly dominant equilibrium. the iteratively not strongly dominated equilibrium and the Nash equilibria of the game Example ⎛ ⎝ 6; 1 3; 1 1; 5 2; 4 4; 2 2; 3 5; 1 6; 1 5; 2 ⎞ ⎠ Answer:

1

No player has as strictly dominant strategy, thus the game does not have a strictly dominant equilibrium.

2

The procedure of iterative elimination of strongly dominated strategies gives (5, 2). Thus the game has an iteratively not strongly dominated equilibrium: (3, 3).

3

The game has one nash equilibrium: (3, 3).

Microeconomics Game Theory

Solution concepts

Theorem

• a. Each strictly dominant equilibrium is an iteratively not

strongly dominated equilibrium. And if the game is finite:

• b. Each Nash equilibrium is an iteratively not strongly

dominated multi-stategy. (So each nash equilibrium survives the procedure.)

• c. An iteratively not strongly dominated equilibrium is a

unique nash equilibrium. Proof.

• 1. Already in first steps of procedure all strategies are removed

with the exception of strictly dominant ones. 2, 3. One verifies that in each step of the procedure the set of nash equilibria remains the same. (See the text book.)

Microeconomics Game Theory

Mixed strategies

Some games do not have a nash equilibrium. Mixed strategy of player i: probability density over X i. With mixed strategies, payoffs have the interpretation of expected payoffs. Nash equilibrium in mixed strategies. Remark: each nash equilibrium is a nash equilibrium in mixed strategies. (See text book for formal proof.)

Microeconomics Game Theory

Bi-matrix-game with mixed strategies

Consider a 2 × 2 bi-matrix-game (A; B) Strategies: (p, 1 − p) for player 1 and (q, 1 − q) for player B. Expected payoffs: f A(p, q) = (p, 1 − p) ∗ A ∗

• q

1 − q

• ,

f B(p, q) = (p, 1 − p) ∗ B ∗

• q

1 − q

• .

Microeconomics Game Theory

Example

Example Determine the nash equilibria in mixed strategies for

• 0; 0

1; −1 2; −2 −1; 1

• .

Answer: f A(p; q) = (−4q + 2)p + 3q − 1, f B(p; q) = (4p − 3)q + 1 − 2p. This leads to the nash equilibrium p = 3/4, q = 1/2.

Microeconomics Game Theory

Example

Example Determine the nash equilibria in mixed strategies for −1; 1 1; −1 1; −1 −1; 1

• .

Answer: p = q = 1/2.

Microeconomics Game Theory

Existence of nash equilibria

Conditional payoff function of player i: fi as a function of xi, given strategies of the other players. Theorem (Nikaido-Isoda.) Each game in strategic form where

1

each strategy set is a convex compact subset of some Rn,

2

each payoff function is continuous,

3

each conditional payoff function is quasi-concave, has a nash equilibrium. Proof. This is a deep theoretical result. A proof can be based on Brouwer’s fixed point theorem. See text book for the proof of a simpler case (Theorem 7.2., i.e. the next theorem).

Microeconomics Game Theory

Theorem of Nash

Theorem Each bi-matrix-game has a nash equilibrium in mixed strategies. Proof. Apply the Nikaido-Isoda result.

Microeconomics Game Theory

Antagonistic game

Consider an antagonistic game: two players and f1 + f2 = 0, i.e. a zero-sum game. Theorem If (a1, a2) and (b1, b2) are nash equilibria, then f1(a1, a2) = f1(b1, b2) and f2(a1, a2) = f2(b1, b2). Proof. f1(a1, a2) ≥ f1(b1, a2) = −f2(b1, a2) ≥ −f2(b1, b2) = f1(b1, b2). In the same way f1(b1, b2) ≥ f1(a1, a2). Therefore f1(a1, a2) = f1(b1, b2) and thus f2(a1, a2) = f2(b1, b2).

Microeconomics Game Theory

Appetizer

t-t-t chess 8 × 8 checkers hex value draw not known draw 1

• pt. strat.

known not known known not known Value: outcome of the game in the case of two rational intelligent players. Optimal strategy for a player: a strategy that guarantees this player at least the value.

Microeconomics Game Theory

Hex

1

Invented independently by Piet Hein and John Nash.

2

http://www.lutanho.net/play/hex.html.

3

Hex can not end in a draw. (’Equivalent’ with Brouwer’s fixed point theorem in two dimensions.)

4

If You can give a winning strategy for hex, then You solved a ’1-million-dollar problem’.

Microeconomics Game Theory

Games in extensive form

Our setting is always non-cooperative with complete information (and for the moment) perfect information and no chance moves. Game tree: Nodes (or histories): end nodes, decision nodes, unique initial node. Directed branches. Payoffs at endnodes. Each non-initial node has exactly one predecessor. No path in tree connects a node with itself. Game is finite (i.e. a finite number of branches and nodes). Actual moves can be denoted by arrows.

Microeconomics Game Theory

Perfect information (ctd.)

Theoretically: Imperfect information can be dealt with by using information sets. The information sets form a partition of the decision nodes. (Example: Figure 7.10.) Perfect information: all information sets are singletons. Solution concept: Nash equilibrium. Games in strategic form are games with imperfect information.

Microeconomics Game Theory

Normalisation

Strategy: specification at each decision node how to move. (This may be much more than a completely elaborated plan of play.) Normalisation: make out (in natural way) of game in extensive form a game in strategic form. So normalisation destroys the perfect information. All terminology and results for games in strategic form now also applies to games in extensive forms.

Microeconomics Game Theory

Solving from the end to the beginning

Example Consider the following game between two (rational and intelligent) players. There is a pillow with 100 matches. They alternately remove 1, 3 or 4 matches from it. (Player 1 begins.) The player who makes the last move wins. Who will win? Answer: the loosing positions are 0, 2, 7, 9, 14, 16, 21, . . ., i.e. the numbers that have remainder 0 or 2 when divided by 7. Because 100/7 has remainder 2, 100 is a loosing position and player 2 has a winning strategy.

Microeconomics Game Theory

Procedure of backward induction (explained at the blackboard) leads to a non-empty set of backward induction multi-strategies. Theorem (Kuhn.) Each backward induction multi-strategy of a finite game in extensive form with perfect information is a nash equilibrium. Proof. See text book. But a nash equilibrium not necessarily is a backward induction multi-strategy.

Microeconomics Game Theory

Subgame perfection

Subgame: game starts at a decision node. Subgame perfect nash equilibrium: a nash equilibrium that remains for each subgame a nash equilibrium. Theorem For every finite extensive form game with perfect information the set of backward induction multi-strategies coincides with the set of subgame perfect nash equilibria. Proof. See text book.

Microeconomics Game Theory

Hex-game revisited

As the game cannot end in a draw, general theory guarantees that player 1 or player 2 has a winning strategy. Here is a proof that player 1 has such a strategy by a strategy-stealing argument:

Microeconomics Game Theory

Hex-game revisited (ctd.)

suppose that the second player has a winning strategy, which we will call S. We can convert S into a winning strategy for the first player. The first player should make his first move at random; thereafter he should pretend to be the second player, "stealing" the second player’s strategy S, and follow strategy S, which by hypothesis will result in a victory for him. If strategy S calls for him to move in the hexagon that he chose at random for his first move, he should choose at random again. This will not interfere with the execution of S, and this strategy is always at least as good as S since having an extra marked square on the board is never a disadvantage in hex.

Microeconomics Game Theory

Games in extensive form: extensions

Three extensions: Imperfect information. Incomplete information: the solution concept here is that of Bayesian equilibrium (7.2.3.). [Next part of course.] Randomization.

Microeconomics Game Theory

Imperfect information

Imperfect information. Can be dealt with by using information sets. The information sets form a partition of the decision nodes. (Example: Figure 7.10.) Perfect information: all information sets are singletons. Strategy: specification at each information set how to move. The procedure of backward induction cannot be applied anymore, but the notion of subgame perfect Nash equilibria still makes sense (when ’subgame’ is properly defined). [Next part of course.] Subgame: not all decision nodes define anymore a

• subgame. (Example: Figure 7.20.) [Next part of course.]

Nash equilibria need not always exist. (Example: Figure 7.23.) [Next part of course.]

Microeconomics Game Theory

Randomization

Three types of strategies: pure, mixed and behavioural strategies.

Microeconomics Game Theory

Randomization

[Next part of the course.] A pure strategy of player i is a book with instructions where there is for each decision node for i a page with the content which move to make at that node. So the set of all pure strategies of player i is a library of such books. A mixed strategy of player i is a probability density on his

• library. Playing a mixed strategy now comes down to

choosing a book from this library by using a chance device with the prescribed probability density.

Microeconomics Game Theory

Randomization (ctd.)

A behavioural strategy, is like a pure strategy also a book, but of a different kind. Each page in the book still refers to a decision node, but now the content is not which move to make but a probability density between the possible moves. For many games (for instance those with perfect recall) it makes no difference whatever if players employ mixed or behavioural strategies.

Microeconomics Game Theory

Nash

John Nash (1928 – 2015).

• Mathematician. (Economist ?)

Nobel price for economics in 1994, together with Harsanyi and Selten. Abel Price for mathematics in 2015. Just after having received it he was killed in a car crash. Got this price for his PhD dissertation (27 pages) in 1950. http://topdocumentaryfilms.com/a-brilliant-madne .