ECON 100 Tutorial: Week 9 www.lancaster.ac.uk/postgrad/murphys4/ [email protected] office: LUMS C85 Question 1(a) Define a dominant strategy equilibrium – One strategy dominates another strategy when it yields a higher payoff regardless of the strategy of the other player or players – A dominant strategy is one which dominates all other strategies for a particular player. (i.e. your best response function is the same regardless of what the other person does) – A dominant strategy equilibrium, if it exists, is the combination of strategies and payoffs if each player plays a dominant strategy. • This has weak and strong forms • A strategy weakly dominates another strategy when it yields payoffs which are as high or higher than the other regardless of the strategies of the other player. Question 1(a) • Example (player 1 is blue): Question 1(b) Define a Nash equilibrium – A best response function • A function which returns the strategy with the highest payoff given the action of the other player – A mutual best response • When all players play best response strategies – A Nash Equilibrium is a situation where economic actors each choose their best strategy given the strategies that all other actors have chosen (so in short, Nash Equilibrium is a mutual best response) – In other words, a Nash equilibrium is a set of strategies, one for each player, such that no player has incentive to change his or her strategy given what the other players are doing. Question 1(b) Example: Lecture 20 slide 4: Isoprofit Curves and RFs (2) Assume the strategic variable is output (this is a Cournot competitition game): q2 q2 RF1 RF1 q2 0 RF2 q1‘ q1 q1’’ q1 0 RF is a best response function. So each firm’s strategy in this case their best response function. The Nash equilibrium is where both firms play their RF quantity (where the to best response functions cross each other). q1 Question 1(b) • Example (player 1 is blue): Question 1(c) Define a subgame perfect Nash equilibrium • A game is represented in extensive form if it is shown as a sequence of decisions (often in a tree diagram) • A node in an extensive form game has perfect information if all players know which decision was made in all nodes which precede that node • A subgame is a set of nodes (decisions for some player) and payoffs that follow a perfect information node • A subgame perfect Nash equilibrium is a NE such that players’ strategies constitute a NE in every subgame of the original game. – It may be found by backward induction. First, one determines the optimal strategy of the player who makes the last move of the game. Then, the optimal action of the next-to-last moving player is determined taking the last player's action as given. The process continues in this way backwards in time until all players' actions have been determined. – Thus it is a refinement of NE, and NE may exist which are not subgame perfect Question 1(c) • Example Question 1(ii) Distinguish carefully between the three concepts above. • I think the example in the previous slide does a good job of distinguishing and illustrating these three concepts. • In the previous example, there is no dominant strategy equilibrium. • Is there a NE which is not subgame perfect? – [(g),(ns,s)] is a NE which is subgame perfect – [(ng),(s,s)] is a NE which is not subgame perfect Question 2(a) Using diagrams where appropriate, explain what is meant by Quantity reaction functions. • If the strategies are quantities, a quantity reaction function is a best response function where the choice of quantity for one player is the best response to the choice of quantity for the other Question 1(b) Example: Lecture 20 slide 4: Isoprofit Curves and RFs (2) Assume the strategic variable is output: q2 q2 RF1 RF1 q2 0 RF2 q1‘ q1 q1’’ q1 0 q1 RF is a best response function. The Nash equilibrium is where bother firms play their RF quantity. Question 2(b) Using diagrams where appropriate, explain what is meant by Cournot competition. Assumptions: • N players - firm 1, firm 2, … firm n • Firms choose output strategies simultaneously • Zero conjectural variations • Both firms have complete information • The firms produce a homogeneous good • The firms face the same costs The previous example is a 2 player Cournot competition problem Question 2(c) Using diagrams where appropriate, explain what is meant by Bertrand competition. Lecture 20 slide 7 & 8 discuss Bertrand competition Assumptions: • N players - firm 1, firm 2, … firm n • Firms choose price strategies simultaneously • Zero conjectural variations • Both firms have complete information • The firms produce a homogeneous good • The firms face the same costs Question 2(c) Assume MC constant, MC = AC. The 45° line represents where firm1’s price = firm2’s price. Let’s first consider Player 1’s best response function. • If P2 chooses price less than MC, P1 will still choose price = MC (=AR). • If P2 choses P>MC, P1 can choose P just a bit lower and capture the entire market (but still above MC). • If P2 chooses P>monopoly price, P1 chooses P=monopoly price, since P1 will capture entire market and this P maximizes profit. Question 2(c) • Player 2’s BR is similar • Equilibrium is P = MC, and both players split the market. Note: the important thing to remember is that in a Bertrand competition problem, P = MC. Question 2(c) Lecture 20 slide 8 presents a similar picture of Bertrand Competition Assumptions: • 2 players - firm 1, firm 2 • Firms choose price strategies simultaneously Extension: • Zero conjectural variations N firm oligopoly • Both firms have complete information • The firms produce a homogeneous good • The firms face the same costs p2 RF1 Bertrand-Nash equilibrium RF2 Firm 1: D p 1 if p 1 p 2 1 D 1 p 1 , p 2 D p 1 if p 1 p 2 2 0 if p 1 p 2 p 1 p 2 mc mc p1 Question 3 Develop a two player normal form game (this can be, but does not have to be Economics related – use your imagination). Select the strategy choices of the players and present the resulting payoffs in a normal form game payoff matrix. Identify any dominant strategies of the players, and identify any Nash equilibria. Be prepared to show your game and the equilibrium outcomes to your class. Question 4 Develop a two player extensive form game (this can be, but does not have to be Economics related – use your imagination. It can but does not need to be related to the game created in Question 3). Decide the number of periods, indicate the strategy choices of the players and the resulting payoffs. Identify the Nash equilibria of each subgame and the subgame perfect Nash equilibrium. Be prepared to show your game and the equilibrium outcomes to your class. Question 5 Suppose your only wealth is in an asset which is worth 144 with probability 2/3 and worth 225 with probability 1/3. Your utility function is U=Wealth What is your expected utility? Suppose you could insure against this risk – what is the maximum insurance premium (P*) that you would be prepared to pay to eliminate this risk? Question 5 Suppose your only wealth is in an asset which is worth 144 with probability 2/3 and worth 225 with probability 1/3. From Ian’s slides, we know how to find expected value: = Σ = 1 1 + 2 2 So the expected worth of this asset is: EW = 144 * probability asset is worth 144 + 225 * probability asset is worth 225 EW = (144 x 2/3) + (225 x 1/3) EW = 96 + 75 EW = 171 Question 5 Your utility function is U=Wealth. What is your expected utility? From Ian’s slides, we know how to find Expected Utility: = Σ ( ) = 1 (1 ) + 2 (2 ) We can apply that here: EU = the probability the asset is worth 144 * utility of 144 + the probability the asset is worth 225 * utility of 225 EU = (2/3)(U(144)) + (1/3) (U(225)) EU = (2/3)( (144)) + (1/3)( (225)) EU = (2/3)(12) + (1/3)(15) EU = 8 + 5 EU = 13 Question 5 Suppose you could insure against this risk – what is the maximum insurance premium (P*) that you would be prepared to pay to eliminate this risk? We know that the Expected Utility of our risky asset is EU = 13. However, the utility of a risk-free asset with the same expected worth as our asset is greater: U(EW) = U(171) = 171 = 13.077 Which is greater than EU = 13 The insurance premium is the price (P*) we would pay to hold a risk-free asset which always returns a value whose utility is the expected utility of the risky asset (EU = 13). To find this, we will set U(EW – P*) = EU and solve for P*: U(EW-P*) = EU U(171-P*) = 13 (171-P*) = 13 171 – P* = 169 P* = 2

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