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### Communication Networks

A Second Course

Jean Walrand

Department of EECS

University of California at Berkeley

Concave, Learning, Cooperative

- Concave Games
- Learning in Games
- Cooperative Games

Concave Games

Motivation

- In many applications, the possible actions belong to a continuous set.For instance one chooses prices, transmission rates, or power levels.
- In such situations, one specifies reward functions instead of a matrix or rewards.
- We explain results on Nash equilibria for such games

Concave Games: Preliminaries

Many situations are possible:

3 NE

1 NE

No NE

J.B. Rosen, “ Existence and Uniqueness of Equilibrium Points forConcave N-Person Games,” Econometrica, 33, 520-534, July 1965

Concave Game

Specialized Case:

Concave Game

Definition: Diagonally Strictly Concave

Concave Game

Theorem: Uniqueness

Concave Game

Theorem: Uniqueness - Bilinear Case:

Concave Game

Local Improvements

Learning in Games

Motivation

Examples

Models

Fictitious Play

Stochastic Fictitious Play

Fudenberg D. and D.K. Levine (1998), The Theory of Learning in Games. MIT Press, Cambridge, Massachusetts. Chapters 1, 2, 4.

Motivation

Explain equilibrium as result of players “learning” over time (instead of as the outcome of fully rational players with complete information)

Examples: 1

Fixed Player Model

If P1 is patient and knows P2 chooses her play based on her forecast of P1’s plays, then P1 should always play U to lead P2 to play R

A sophisticated and patient player who faces a naïve opponent can develop a reputation for playing a fixed strategy and obtain the rewards of a Stackelberg leader

Large Population Models

Most of the theory avoids possibility above by assuming random pairings in a large population of anonymous users

In such a situation, P1 cannot really teach much to the rest of the population, so that myopic play (D, L) is optimal

Naïve play: Ignore that you affect other players’ strategies

Examples: 2

Cournot Adjustment Model

Each player selects best response to other player’s strategy in previous period

Converges to unique NE in this case

This adjustment is a bit naïve …

2

BR1

BR2

NE

1

Models

Learning Model: Specifies rules of individual players and examines their interactions in repeated game

Usually: Same game is repeated (some work on learning from similar games)

Fictitious Play: Players observe result of their own match, play best response to the historical frequency of play

Partial Best-Response Dynamics: In each period, a fixed fraction of the population switches to a best response to the aggregate statistics from the previous period

Replicator Dynamics: Share of population using each strategy grows at a rate proportional to that strategy’s current payoff

Fictitious Play

Each player computes the frequency of the actions of the other players (with initial weights)

Each player selects best response to the empirical distribution (need not be product)

Theorem:

Strict NE are absorbing for FP

If s is a pure strategy and is steady-state for FP, then s = NE

Proof: Assume s(t) = s = strict NE. Then, with a := a(t) …, p(t+1) = (1 – a)p(t) + ad(s), so that u(t+1, r) = (1 – a)u(p(t), r) + au(d(s), r),which is maximized by r = s if u(p(t), r) is maximized by r = s.

Converse: If converges, this means players do not want to deviate, so limit must be NE…

Fictitious Play

Assume initial weights (1.5, 2) and (2, 1.5). Then(T, T) (1.5, 3), (2, 2.5) (T, H), (T, H) (H, H), (H, H), (H, H) (H, T)…

Theorem:

If under FP empirical converge, then product converges to NE

Proof: If strategies converge, this means players do not want to deviate, so limit must be NE…

Theorem:

Under FP, empirical converge if one of the following holds

2x2 with generic payoffs

Zero-sum

Solvable by iterated strict dominance

…

Note: Empirical distributions need not converge

Fictitious Play

Assume initial weights (1, 20.5) for P1 and P2. Then(A, A) (2, 20.5) (B, B) (A, A) (B, B) (A, A), etc

Empirical frequencies converge to NE

However, players get 0

Correlated strategies, not independent(Fix: Randomize …)

Stochastic Fictitious Play

Motivation:

Avoid discontinuity in FP

Hope for a stronger form of convergence: not only of the marginals, but also of the intended plays

Stochastic Fictitious Play

Definitions:

Reward of i = u(i, s) + n(i, si), n has positive support on interval

BR(i, s)(si) = P[n(i, si) is s.t. si = BR to s]

Nash Distribution: if si = BR(i, s), all i

Harsanyi’s Purification Theorem:

For generic payoffs, ND NE if support of perturbation 0.

Key feature: BR is continuous and close to original BR.

Matching Pennies

Stochastic Fictitious Play

Theorem (Fudenberg and Kreps, 93):

Assume 2x2 game has unique mixed NE

If smoothing is small enough, then NE is globally stable for SFP

Theorem (K&Y 95, B&H 96)

Assume 2x2 game has unique strict NE

The unique intersection of smoothed BR is a global attractor for SFPAssume 2x2 game has 2 strict NE and one mixed NE. The SFP converges to one of the strict NE, w.p. 1.

Note: Cycling is possible for SFP in multi-player games

Stochastic Fictitious Play

Other justification for randomization: Protection against opponent’s mistakes

Learning rules should be

Safe: average utility ≥ minmax

Universally consistent: utility ≥ utility if frequency were known but not order of plays

Randomization can achieve universal consistency (e.g., SFP)

Stochastic Fictitious Play

Stimulus-Response (Reinforcement learning):

Increase probability of plays that give good results

General observation: It is difficult to discriminate learning models on the basis of experimental data: SFP, SR, etc. seem all about comparable

Cooperative Games

- Motivation
- Notions of Equilibrium
- Nash Bargaining Equilibrium
- Shapley Value

Cooperative Games: Motivation

- The Nash equilibriums may not be the most desirable outcome for the players.
- Typically, players benefit by cooperating.
- We explore some notions of equilibrium that players achieve under cooperation.

Fixed Point Theorems

Theorem (Brower):

Brower

Labels (1, 3)

Labels (2, 3)

One path through

doors (1, 2) mustend up in triangle

(1, 2, 3).[Indeed: Odd numberof boundary doors.]

Brower

Take small triangle (1, 2, 3)

Divide it into triangles as before; it contains another (1, 2, 3);

Continue in this way.

Pick z(n) in triangle n. Let z = lim z(n).

Claim: f(z) = z.

Proof: If f(z) is not z, then z(n) and f(z(n)) are in different small

triangles at stage n; but then z(n) cannot be in a (1, 2, 3)

triangle ….

Notes on Kakutani

Theorem:

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