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Communication Networks

Communication Networks. A Second Course. Jean Walrand Department of EECS University of California at Berkeley. Games: Non-Cooperative. One-Shot (Static) Game Nash Equilibrium Cournot Leader-Follower (Stackelberg) Bayes-Cournot Principal-Agent Problem Existence of NE. Static Game.

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Communication Networks

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  1. Communication Networks A Second Course Jean Walrand Department of EECS University of California at Berkeley

  2. Games: Non-Cooperative • One-Shot (Static) Game • Nash Equilibrium • Cournot • Leader-Follower (Stackelberg) • Bayes-Cournot • Principal-Agent Problem • Existence of NE

  3. Static Game Actions of Bob Reward of Bob Actions of Alice Reward of Alice • Example: Matching pennies • One-Shot Game • Both players play simultaneously • Each player knows both reward functions

  4. Static Game Actions of Bob Actions of Alice • Game Example: 2x2 • One-Shot Game • Both players play simultaneously • Each player knows both reward functions

  5. Static Game • Generally: Reward of player i: Strategy of player i: Goal of player i: • One-shot game • All players play simultaneously, independently • Each player knows all the reward functions

  6. Static Game • Example: Matching pennies

  7. Static Game • Nash Equilibrium In the non-randomized case: In words: No player has an incentive to deviate unilaterally.

  8. Static Game • Example: Matching pennies

  9. Static Game • Recall: Cournot Duopoly Two firms produce quantity q1 and q2 of a product The price is A – q1 – q2 For i = 1, 2 the profit of firm i is qi(A – q1 – q2) - Cqi • One-shot game • Continuous action space • Each player knows both reward functions

  10. Static Game • Nash Equilibrium: Cournot Recall: Profit of firm 2 is q2(A – q1 – q2 – C) This is maximized for q2 = (A – q1 – C)/2 =: (B – q1)/2 Thus, qi = B/3 and reward is ui = B2/9.

  11. Static Game • Cournot: Cooperation u2 = q2(A – q1 – q2 – C) Assume firms cooperate and split the revenues Then they choose qi = q/2 where q maximizes u1 + u2 = 2ui = q(A – q – C) = q(B – q) The maximum is q = B/2, so that qi = B/4 and ui = B2/8 The firms produce less and make more profit when they cooperate than when they compete.

  12. Static Game • Stackelberg: Cournot Recall: u2 = q2(B – q1 – q2) Assume firm 1 announces q1 and firm 2 follows Then q2 = (B – q1)/2, so that u1 = q1(B - q1)/2 This is maximized by q1 = B/2 so that q2 = B/4 Hence u1 = B2/8 and u2 = B2/16 Intuition: Leader (1) has advantage.

  13. Static Game • Nash Equilibrium: Bayes Cournot Recall: Profit of firm 2 is u2 = q2(A – q1 – q2 – C2) Assume firm i knows Ci and the distribution of C2 - i u2 is maximized for q2 = (A – q1 – C2)/2 Similarly, q1 = (A – q2 – C1)/2 However, no player can solve…. Assume P2 tries to maximize E[u2 | q2, C2]over q2 E[u2 | q2, C2] = q2(A – Q1 – q2 – C2) where Q1 = E(q1) This is maximized for q2 = (A – Q1 – C2)/2 So, Q2 = (B – Q1)/2 with B = A – C, C = E(C2) = E(C1). Hence, Q1 = Q2 = Q = B/3, as before. This gives q2 = (A – B/3 – C2)/2 and q1 = (A – B/3 – C1)/2 Then u2 = (A – B/3 – C2)(B/3 + C1/2 – C2/2)/2 Thus, E(u2) = B2/9 + s2/4 where s2 := var(Ci) Intuition: Produce more when cost is small ….

  14. Static Game • Nash Equilibrium: Bayes Cournot vs. Full Information Assume costs Ci are random and independent with C = E(Ci), s2 = var(Ci) However, the costs are known by both firms Recall: u2 = q2(A – q1 – q2 – C2) This is maximized for q2 = (A – q1 – C2)/2 Similarly, q1 = (A – q2 – C1)/2 Solving, we get q1 = (A + C2 – 2C1)/3 and q2 = … Hence, u2 = (A + C1 – 2C2)2/9 so that E(u2) = B2/9 + 5s2/9 where B = A – C In the incomplete information case E(u2) = B2/9 + s2/4 Thus, the cost of lack of information is 11s2/36

  15. Static Game • Stackelberg: Bayes Cournot Recall: u2 = q2(A – q1 – q2 – C2) Assume firm 1 announces q1 Then q2 = (A – q1 – C2)/2 Hence, u1 = q1(A – q1 - q2 – C1) = q1(A – q1 + C2 – 2C1)/2 Note that E[u1 | q1, C1] = q1(A – q1 + C – 2C1)/2 This is maximized by q1 = (A + C – 2C1)/2 So, u1 = [B + 2(C – C1)][B + 2(C2 – C1)]/8 Thus, E(u1) = B2/8 + s2/2 and E(u2) = B2/16 + s2/2 Intuition: Produce more when cost is small …. If costs are known, E(u1) = B2/8 + 5s2/8 and E(u2) = B2/16 + 13s2/16 The cost of lack of information is s2/8 to 1 and 5s2/16

  16. Static Game • Cournot: Summary

  17. Principal-Agent Problem Menu of Contracts: (t(q), q(q)), for q in Q Principal Agent Type q Declares a type q ’ • Basic Model Principal’s Utility: S(q) – t S’(.) > 0, S”(.) < 0 S(0) = 0 Agent’s Utility: t – F - qq q = agent’s “type” = Marginal prod. cost Agent can be of two types: Efficient qL or Inefficient qH > qL. Principal does not know the type! Must design good contract.

  18. Principal-Agent Problem • Contract: • Bayesian Optimal Contract:

  19. Principal-Agent Problem • Bayesian Optimal Contract: UL is the rent that the efficient agent can get by mimicking the inefficient agent. Thus, tradeoff between efficiency and information rent.

  20. Principal-Agent Problem q Principal Agent (t(q), q(q)) q Principal Agent (T(m(q)), Q(m(q))) • Revelation Principle

  21. Existence of Nash Equilibrium • Theorem (Nash) Every finite static game has at least one Nash equilibrium (possibly randomized)

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