IPM Microeconomics-2 Past Paper PDF 2024

IPM Microeconomics- 2 2024 Session 10 and 11 Joysankar Bhattacharya Games Any situation in which individuals must make strategic choices and in which the final outcome will depend on what each person chooses to do can be viewed as a game. A Nash equilibrium is a pair of str...

IPM Microeconomics- 2 2024 Session 10 and 11 Joysankar Bhattacharya Games Any situation in which individuals must make strategic choices and in which the final outcome will depend on what each person chooses to do can be viewed as a game. A Nash equilibrium is a pair of strategies (a*,b*) in a two- player game such that a* is an optimal strategy for A against b* and b* is an optimal strategy for B against a*. Players can not benefit by deviating from equilibrium. Self-reinforcing Not every game has a Nash equilibrium, and some games may have multiple Nash equilibria. Dominant Strategies and Nash Equilibria A dominant strategy is a strategy that is better than any other strategy that a player might choose, no matter what strategy the other player follows. Note: When a player has a dominant strategy, that strategy will be the player's Nash Equilibrium strategy !! A dominant strategy is optimal regardless of the strategy adopted by an opponent. A player has a dominated strategy when the player has another strategy that gives it a higher payoff no matter what the other player does. Advertising Game Two firms (A and B) must decide how much to spend on advertising Each firm may adopt either a higher (H) budget or a low (L) budget. the dominant strategy for B is L since this yields a larger payoff regardless of A’s choice. A will recognize that B has a dominant strategy and choose the strategy which will yield the highest payoff, given B’s choice of L. The strategy choice will be (A: L, B: L)  Dominant or Dominated Strategy Why look for dominant or dominated strategies? A dominant strategy equilibrium is particularly compelling as a "likely" outcome. Similarly, because dominated strategies are unlikely to be played, these strategies can be eliminated from consideration in more complex games. This can make solving the game easier. 5  Dominated Strategy Toyota Honda "Build Large" is dominated for each player By eliminating the dominated strategies, we can reduce the game !! 6 Rock-Scissors-Paper Game Rock-Scissors-Paper Game Each player simultaneously forms his or her hand into the shape of either a rock, a piece of paper, or a pair of scissors Rule: rock beats (blunts) scissors, scissors beats (cuts) paper, and paper beats (wraps) rock The zero payoffs along the diagonal show that if players adopt the same strategy, no payments are made. In other cases, the payoffs indicate a $1 payment from the loser to winner under the usual hierarchy Rock-Scissors-Paper Game : No Nash !!! This game has no Nash equilibrium. Any strategy pair is unstable since it offers at least one of the players an incentive to adopt another strategy. For example, (A: Scissors, B: Scissors) provides and incentive for either A or B to choose Rock. Also, (A: Paper, B: Rock) encourages B to choose Scissors. Stag-Hare Game : Multiple Nash Two Hunters : A or B Two Strategies: Stag or Hare Multiple Nash : (Stag, stag) & (Hare, Hare) A Threat Game in Advertising The adoption of strategy H by firm A has disastrous consequences for B (-50 if L is chosen, -25 if H is chosen). Without communication, the Nash equilibrium is (A: L, B: H) which results in profits of +15 for A and +10 for B. However, A might threaten to use strategy H unless B plays L to increase profits by 5. If a game is replayed many times, cooperative behavior my be fostered. Some market are thought to be characterized by “tacit collusion” although firms never meet.  Game Theory (The Prisoners’ Dilemma) Game theory is a tool for studying strategic behavior (takes into account the expected behavior of others and the mutual recognition of interdependence). Rules: The rules describe the setting of the game, the actions the players may take, and the consequences of those actions. In the prisoners’ dilemma game, two prisoners (Art and Bob) have been caught committing a petty crime. Each is held in a separate cell and cannot communicate with each other. 12 The Prisoners’ Dilemma : No Communication The Prisoners’ Dilemma Strategies Strategies are all the possible actions of each player. Art and Bob each have two possible actions: 1. Confess to the larger crime. 2. Deny having committed the larger crime. With two players and two actions for each player, there are four possible outcomes: 1. Both confess. 2. Both deny. 3. Art confesses and Bob denies. 4. Bob confesses and Art denies. 15 Payoffs Each prisoner can work out what happens to him can work out his payoff in each of the four possible outcomes. payoff matrix. A payoff matrix is a table that shows the payoffs for every possible action by each player for every possible action by the other player. 16 17 Outcome If a player makes a rational choice in pursuit of his own best interest, he chooses the action that is best for him, given any action taken by the other player. If both players are rational and choose their actions in this way, the outcome is an equilibrium called Nash equilibrium—first proposed by John Nash. 18 Bob’s view of the world 19 Bob’s view of the world 20 Art’s view of the world 21 Art’s view of the world 22 Equilibrium 23 Collusion Suppose that the two firms enter into a collusive agreement. A collusive agreement is an agreement between two (or more) firms to restrict output, raise the price, and increase profits. Such agreements are illegal in the some countries and are undertaken in secret. Firms in a collusive agreement operate a cartel. 24 The strategies that firms (Trick and Gear) in a cartel can pursue are to ▪ Comply ▪ Cheat Because each firm has two strategies, there are four possible combinations of actions for the firms: 1. Both comply. 2. Both cheat. 3. Trick complies and Gear cheats. 4. Gear complies and Trick cheats. 25 Possible Outcomes ▪ If both comply, each firm makes $2 million a week. ▪ If both cheat, each firm makes zero economic profit. ▪ If Trick complies and Gear cheats, Trick incurs an economic loss of $1 million and Gear makes an economic profit of $4.5 million. ▪ If Gear complies and Trick cheats, Gear incurs an economic loss of $1 million and Trick makes an economic profit of $4.5 million. 26 Payoff Matrix 27 Trick’s view of the world 28 Trick’s view of the world 29 Gear’s view of the world 30 Gear’s view of the world 31 Equilibrium 32 Nash Equilibrium The Nash equilibrium is that both firms cheat. The quantity and price are those of a competitive market, and the firms make zero economic profit. 33  Sequential Move Games Games in which one player (the first mover) takes an action before another player (the second mover). The second mover observes the action taken by the first mover before deciding what action it should take. 34 Sequential Move Game Players, when makes moves, have to consider what the opponents may do. Game Trees are commonly used to specify all possible moves by all players and all possible outcome and payoffs. Nodes : Places where players make moves. - Root - Terminal nodes Branches Possible choices of players Backward induction is a procedure for solving a sequential- move game by starting at the end of the game tree and finding the optimal decision for the player at each decision point. Tic-Tac-Toe ( 362,880 terminal nodes !!) Sequential Move : Extensive Form Games A game can have complex temporal structure. Information set of players who moves when and under what circumstances what actions are available when called upon to move what is known when called upon to move what payoffs each player receives Foundation is a game tree Big Monkey-Little Monkey Game Big Monkey and Little Monkey eat fruit F, which dangle from the extreme tip of a lofty branch of the tree. The tree produces only one fruit. To get it, at least one of the monkeys must climb the tree and shake the branch bearing the fruit until the fruit comes loose and falls to the ground. The fruit is worth 10 calories of energy. Climbing the tree uses 2 calories for Big Monkey, but uses no energy for Little Monkey, who is smaller. If Little Monkey climbs the tree and shakes it down, Big Monkey will eat 90% of the fruit (or 9 calories) before Little Monkey climbs back down, and Little Monkey will get only 10% of the fruit (or 1 calorie). If Big Monkey climbs the tree and Little Monkey waits, Little Monkey will get 40% of the fruit and Big Monkey will get 60%. If both monkeys climb the tree, Big Monkey will get 70% of the fruit and Little Monkey will get 30%. Assume each monkey is simply interested in maximizing his caloric intake. Fundamental Tools Big Monkey (BM) – Little Monkey (LM) Fruit(F) from the tree (only one per tree) = 10 Calories Climb the tree to get the fruit Cost to get the fruit : 2 Calories for Big Monkey zero for Little Monkey Payoff : Both climb : BM 5 Calories – LM 3 Calories BM climbs : BM 4 Calories – LM 4 Calories LM climbs : BM 9 Calories – LM 1 Calories Sequential Moves : Two Possibilities BM decides first what to do LM decides first what to do Big Monkey-Little Monkey Game Big Monkey decides First BM decides first (BM=w, LM=c): Big Monkey w c Little Monkey Little Monkey w c w c 0,0 9,1 4,4 5,3 41 Little Monkey decides First LM decides first : (BM= c , LM = w) Little Monkey w c Big Monkey Big Monkey w c w c 0,0 4,4 1,9 3,5 42 Sequential Move Games – Game Tree 43 The Predation Game Nasty Guys is an incumbent firm producing bricks. SIC (Sweet Innocent Corporation) is a potential new entrant in the brick market. Nasty Guys says that if SIC enters then it will “squish them like a bug”. What should SIC do? The Predation Game Don’t SIC=0, NG=100 enter SIC Fight SIC = -10, NG = -10 NG Enter Don’t fight SIC = 30, NG = 30 45 The Predation Game Don’t SIC=0, NG=100 enter SIC Fight SIC = -10, NG = -10 NG Enter If SIC actually enters, then ‘fighting’ is an incredible Don’t threat – it hurts SIC but also fight SIC = 30, NG = 30 hurts NG. So SIC knows the threat is just bluff !! 46 The Predation Game Don’t SIC=0, NG=100 enter SIC Fight SIC = -10, NG = -10 NG Enter So the equilibrium is: Don’t SIC will enter fight SIC = 30, NG = 30 NG will not fight 47 Credible Commitments When Hernán Cortés arrived in Mexico (1518-20) he ordered that his ships should be burnt !! This seems silly !! His troops were vastly outnumbered Surely it is better to keep an ‘escape route’ home? 48 Cortés @ Mexico (1518-20) Think of Cortes trying to motivate his own soldiers !! Fight Hard C = 100, S = 0 Keep Ships S Be careful C = 0, S = 10 Fight Hard C = 100, S = 0 C S Burn ships Be careful C = -100, S = -100 50 If no retreat possible, will fight hard or die. But if retreat is possible, may fight less hard and ‘run away’ !! Fight Hard C = 100, S = 0 Keep Ships S Be C = 0, S = 10 careful Fight Hard C = 100, S = 0 C S Burn ships Be careful C = -100, S = -100 51 So Cortes wants to burn his ships. It is a credible commitment not to retreat – and this alters how his own troops behave !! Fight Hard C = 100, S = 0 Keep Ships S Be careful C = 0, S = 10 Fight Hard C = 100, S = 0 C S Burn ships Be C = -100, S = -100 careful 52  Mixed Strategy Equilibrium A mixed strategy equilibrium is a pair of mixed strategies that are mutual best responses.  Pure Strategies Player B L R U (1,2) (0,4) Player A D (0,5) (3,2) So the game has no Nash equilibria in pure strategies. Even so, the game does have a Nash equilibrium, but in mixed strategies.  Mixed Strategies Instead of playing purely Up or Down, Player A selects a probability distribution ( , 1 − ), meaning that with probability Player A will play Up and with probability 1 − will play Down. Player A is mixing over the pure strategies Up and Down. The probability distribution ( ,1− ) is a mixed strategy for Player A. 𝑈 𝑈 𝑈 𝑈 𝑈 𝑈 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋  Mixed Strategies Similarly, Player B selects a probability distribution ( ,1 − ), meaning that with probability Player B will play Left and with probability 1 − will play Right. Player B is mixing over the pure strategies Left and Right. The probability distribution ( ,1 − ) is a mixed strategy for Player B. 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋  Mixed Strategies Player B L, R, 1 – U, (1,2) (0,4) Player A D, 1 – (0,5) (3,2) 𝑼 𝑼 𝑳 𝑳 𝝅 𝝅 𝝅 𝝅  Mixed Strategies Player B L, R,1– U, (1,2) (0,4) Player A D,1– (0,5) (3,2) If B plays Left her expected payoff is 𝑼 𝑼 𝑳 𝑳 𝝅 𝝅 𝝅 𝝅  Mixed Strategies Player B L, R,1– U, (1,2) (0,4) Player A D,1– (0,5) (3,2) If B plays Left her expected payoff is If B plays Right her expected payoff is 𝑼 𝑼 𝑳 𝑳 𝝅 𝝅 𝝅 𝝅  Mixed Strategies Player B L, R ,1 – U, (1,2) (0,4) Player A D,1– (0,5) (3,2) If then B would play only Left. 𝑼 𝑳 𝑳 𝑼 𝝅 𝝅 𝝅 𝝅  Mixed Strategies Player B L, R ,1 – U, (1,2) (0,4) Player A D,1– (0,5) (3,2) If then B would play only Right. 𝑼 𝑳 𝑳 𝑼 𝝅 𝝅 𝝅 𝝅  Mixed Strategies Player B L, R ,1 – U, (1,2) (0,4) Player A D,1– (0,5) (3,2) So for there to exist a Nash equilibrium, B must be indifferent between playing Left or Right; i.e. 𝑼 𝑼 𝑳 𝑳 𝝅 𝝅 𝝅 𝝅  Mixed Strategies Player B L, R ,1 – U, (1,2) (0,4) Player A D, − (0,5) (3,2) If A plays Up his expected payoff is 𝑳 𝑳 𝑼 𝑼 𝝅 𝝅 𝝅 𝟏 𝝅  Mixed Strategies Player B L, R ,1 – U, (1,2) (0,4) Player A D, − (0,5) (3,2) If A plays Up his expected payoff is If A plays Down his expected payoff is 𝑳 𝑳 𝑼 𝑼 𝝅 𝝅 𝝅 𝟏 𝝅  Mixed Strategies Player B L, R,1– U, (1,2) (0,4) Player A D, − (0,5) (3,2) If then A would play only Up.. 𝑳 𝑳 𝑼 𝑼 𝝅 𝝅 𝝅 𝟏 𝝅  Mixed Strategies Player B L, R ,1 – U, (1,2) (0,4) Player A D, − (0,5) (3,2) If then A would play only Down 𝑳 𝑳 𝑼 𝑼 𝝅 𝝅 𝝅 𝟏 𝝅  Mixed Strategies Player B L, R,1– U, (1,2) (0,4) Player A D, − (0,5) (3,2) So for there to exist a Nash equilibrium, A must be indifferent between playing Up or Down; i.e. 𝑳 𝑳 𝑼 𝑼 𝝅 𝝅 𝝅 𝟏 𝝅 Mixed Strategies Player B L R U (1,2) (0,4) Player A D (0,5) (3,2) So the game’s only Nash equilibrium has A playing the mixed strategy (3/5, 2/5) and has B playing the mixed strategy (3/4, 1/4). Mixed Strategies Player B L R (1,2) U (0,4) 9/20 Player A D (0,5) (3,2) The payoffs will be (1,2) with probability Mixed Strategies Player B L R (1,2) (0,4) U 9/20 3/20 Player A D (0,5) (3,2) The payoffs will be (0,4) with probability Mixed Strategies Player B L R (1,2) (0,4) U 9/20 3/20 Player A (0,5) D (3,2) 6/20 The pay-off will be (0, 5) with probability Mixed Strategies Player B L R (1,2) (0,4) U 9/20 3/20 Player A (0,5) (3,2) D 6/20 2/20 The payoffs will be (3,2) with probability Mixed Strategies Player B L R (1,2) (0,4) U 9/20 3/20 Player A (0,5) (3,2) D 6/20 2/20 A’s expected Nash equilibrium payoff is B’s expected Nash equilibrium payoff is  Sherlock Holmes and Prof. Moriarty Sherlock Holmes, pursueded by his opponent, Moriarty, leaves London for Dover. The train stops at a station on the way, and he detrains there rather than travelling on to Dover. He has seen Moriarty at the railway station, recognizes that he is very clever and expects that Moriarity will take a faster special train in order to catch him in Dover. Holmes' anticipations turns out to be correct. But what if Moriarity had been still more clever, had estimated Holmes' mental abilities better and had foreseen his actions accordingly? Then, obviously, he would have travelled to the intermediate station [Canterbury]. Holmes again would have had to calculate that, and he himself would have decided to go on to Dover. Whereupon, Moriarity would again have “reacted” di erently. ff  Sherlock Holmes and prof. Moriarty (pure strategy) prof. Moriarty Dover Canterbury Sherlock Dover ( 4 , 25) ( 12 , 5) Holmes Canterbury (16 , 10) (8 , 15)  Sherlock Holmes and prof. Moriarty So no matter how Holmes reasons, Moriarty may predict and catch him !  Sherlock Holmes and prof. Moriarty If myself do not know my destination, then Moriarty cannot guess what I will, do no matter how smart he is Ignorance is a bliss !  Sherlock Holmes and prof. Moriarty ( mixed strategy) prof. Moriarty Dover Canterbury (q) (1–q) Sherlock Dover ( 4 , 25) ( 12 , 5) Holmes (p) Canterbury (16 , 10) (8 , 15) (1–p)  Sherlock Holmes and prof. Moriarty ( mixed strategy) Moriarty chooses q in a way to make Holmes indi erent between going to Dover and Canterbury: 4 q + 12 ( 1 – q ) = 16 q + 8 ( 1 – q) Dover Pay-off Canterbury Pay-off = 𝟒 𝒒 𝟏 ff  Sherlock Holmes and prof. Moriarty ( mixed strategy) Sharlok Holmes chooses p in a way to make Moriarty indi erent between going to Dover and Canterbury: 25 p + 10 ( 1 – p ) = 5 p + 15 ( 1 – p) Dover Pay-off Canterbury Pay-off = 𝟓 𝒑 𝟏 ff  Sherlock Holmes and prof. Moriarty ( mixed strategy) prof. Moriarty Dover Canterbury () () Sherlock Dover ( 4 , 25) ( 12 , 5) Holmes () 1/20 3/20 Canterbury (16 , 10) (8 , 15) () 4/20 12/20  Sherlock Holmes and prof. Moriarty ( mixed strategy) Sharlock Holmes’s expected Nash equilibrium payoff is Prof. Moriarty’s expected Nash equilibrium payoff is