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Unformatted text preview: GAME THEORY Lecture 1516 Sequential Games with Perfect Information Instructor: Nuno Limão slides_414_15_16_2011_no_solutions 1 OUTLINE Motivation Backward Induction and Subgame Perfect Equilibrium Other applications: Competition and 1st mover advantage Other applications: 1st vs. 2nd mover advantage Do People Reason Using Backward Induction? slides_414_15_16_2011_no_solutions 2 MOTIVATION In many situations of strategic interaction at least one player knows what the other(s) did before deciding how to act, so the game is sequential Examples of sequential games Economics: firm entry and production; any bargaining situation with a proposer and accepter (possibly alternating), … Politics: choice of policy platform by challenger after observing incumbent platform, bargaining Sports and games: most, e.g. tennis, chess, … Sequential games are naturally represented by extensive (tree) form and we distinguish between Perfect information (ch 8) so players know exactly what others did (thus each node is a singleton) Imperfect information (ch 9): players do not know exactly what others did in all situations How should we think about solving sequential games? slides_414_15_16_2011_no_solutions 3 Experiment: centipede game Login: veconlab.econ.virginia.edu/login.htm Session: ngl9 slides_414_15_16_2011_no_solutions 4 BACKWARD INDUCTION AND SUBGAME PERFECT EQUILIBRIUM Do we need additional solution concepts to deal with sequential games? Recall: sequential games can be represented by extensive form and have unique strategic form representation. So why not simply use Nash equilibrium? NE may not provide sharp predictions and may include “unreasonable” predictions involving non‐credible strategies or “threats” Employing NE to solve simple sequential games: Recall Ipad example slides_414_15_16_2011_no_solutions 5 Strategic form Firm 2 Firm 1 Ipad apps
Other apps Ipad (1) Ipad (1) Other (1)
Ipad (2) Other (2) Ipad (2) 1,1 [c]
0,0 Other (1)
Other (2) 0,0
1,1 [d] 4 pure strategy NE [a] and [b] are intuitive and seem reasonable: if firm 2 observes x then it chooses x [c] and [d] strategy profiles are also NE since for c firm 1 chooses Ipad and in that case so does firm 2 and thus the strategy that 2 would have chosen if 1 had chosen other does not affect equilibrium payoff. Thus firm 2 is indifferent between Ipad if that choice is “off the equilibrium path”. Does it matter for the outcome of this particular game? No: always end up coordinating. But in other games it can matter (see Ransom game) However, does it seem credible for 2 to “threaten” Ipad if 1 chose other? No, b/c if at Other than 2 always chooses “other”. How should we look for “credible” threats? slides_414_15_16_2011_no_solutions 6 Illustrating unreasonable NE predictions sustained by incredible threats: ransom Setup: Guy deciding whether to kidnap Vivica’s SO slides_414_15_16_2011_no_solutions 7 Strategic form Vivica (kin of victim) has two strategies: Pay ransom or do not pay ransom. Guy (kidnapper) has eight strategies (8=2^3: 2 actions at 3 different info sets) 4 using Do not kidnap (../Kill/Kill, ../Kill/Release, ../Release/Kill, .. /Release/Release). 4 using kidnap (../Kill/Kill, ../Kill/Release, ../Release/Kill, ../Release/Release) What are the pure strategy NE? slides_414_15_16_2011_no_solutions 8 5 Pure strategy NE found using BR approach Kidnap/Release/Kill and pay ransom All 4 strategies where guy does not kidnap and Vivika does not pay ransom Are the NE involving no kidnap reasonable? If Guy believed that Vivika would not pay a ransom then he would be better off not kidnapping (3>2 or 1) If Vivika could convince Guy that she would not pay the ransom then she would want to do so since then he would not kidnap and she would get 5 (>1,3,2,4). Is this threat credible? No, b/c if there is a kidnap then Vivika is better off paying since If no ransom then guy is always better off by killing (so Vivika gets 2) W/ ransom guy is better off releasing and so Vivika gets 3 and guy gets 5 Knowing the above guy would prefer to kidnap, collect ransom and release Reasoning above is known as backward induction and the resulting equilibrium is a subgame perfect Nash equilibrium slides_414_15_16_2011_no_solutions 9 Definitions Subgame: a subset of a sequential game that begins at any non‐terminal node where each player knows all previous actions by others and himself. Substrategy profile: the part of the strategy that prescribes behavior for that subgame (e.g. Kill or release are the two substrategies in each of the 2 last subgames in ransom circled below) slides_414_15_16_2011_no_solutions 10 Definitions (ctd) A strategy profile is a subgame perfect Nash equilibrium if for every subgame its substrategy profile is a NE. Backward induction: method for solving extensive form games for SPNE. It consists of finding the NE for each of the final subgames replacing each of the last subgames with their respective NE payoffs Repeating the procedure until the game is solved Note the key difference between NE and SPNE former requires the actions to be optimal only along the equilibrium path and the latter requires that the actions are optimal even when off the equilibrium path. The requirement of subgame perfection eliminates the Nash equilibrium where Vivica does not pay ransom, because if Guy indeed kidnaps Orlando, Vivica will pay ransom. slides_414_15_16_2011_no_solutions 11 Applying backward induction algorithm to find SPNE in Ransom game Subgame 1 (pay ransom node): guy better by releasing (5>4) so replace w/ (5,3) Subgame 2 (don’t pay ransom): guy better by killing (2>1) so replace w/ (2,2) slides_414_15_16_2011_no_solutions 12 Subgame 3 (kidnap node after applying backward induction): Vivika better off by paying ransom (3>2) so replace with (5,3) Subgame 4 (DNK or K after backward induction): Guy better off kidnapping (5>3) Outcome SPNE slides_414_15_16_2011_no_solutions 13 Properties of SPNE In a game of perfect information there is at least one subgame perfect Nash equilibrium [b/c at each node there is always at least one optimal action] Every subgame perfect NE is a NE but not every NE is a SPNE [b/c SPNE requires the action to be optimal at every node, whether reached in equilibrium or not whereas NE only requires optimality at nodes that are reached in equilibrium] slides_414_15_16_2011_no_solutions 14 Another life and death application: The Cuban missile crisis Setup On October 14, 1962, the US confirmed the presence of Soviet missiles in Cuba. The challenge for President John F. Kennedy was to get the missiles out of Cuba before they became operational, which the CIA estimated would be in about ten days. Otherwise the US would be within range of soviet missiles w/ little warning slides_414_15_16_2011_no_solutions 15 Representing the game The United States had two initial strategies: (1) Blockade the island to prevent Soviet ships in route to Cuba with additional materials, possibly followed by a land invasion (2) Perform an air strike to destroy the missiles before they become operational. If the U.S. went for air strike, then a devastating war between the U.S. and the Soviet Union would break out. If the U.S. went for a blockade, then the Soviet Union would have to decide whether to maintain the missiles or withdraw. If the Soviet Union had maintained the missiles, then the U.S. could go for a blockade or an air strike. slides_414_15_16_2011_no_solutions 16 Extensive form Payoffs reflect that US would prefer USSR to withdraw w/out an airstrike (4>3,2) but if they were maintained then it prefers taking them out (3>1) Exercise: what is the SPNE of the crisis? slides_414_15_16_2011_no_solutions 17 Solution to Cuban crisis … Actual Outcome: US blockaded and USSR withdrew, as predicted by SPNE. slides_414_15_16_2011_no_solutions 18 OTHER APPLICATIONS: COMPETITION AND 1 ST MOVER ADVANTAGES Stackelberg duopoly game setup First one firm, the leader (1), chooses its quantity , q1 Then other firm (2, the follower) observes and chooses q2. Homogeneous goods so industry output is q1 + q2. constant average and marginal cost = 3 Linear demand : p = 19 – 2Q, where Q = market demand=(q1 + q2) in equilibrium so p = 19 – 2(q1 + q2) Stackelberg duopoly game SPNE What is the optimal quantity in the last subgame, i.e. what is q2? 2 = (p – 3) q2 = (16 ‐ 2(q1 + q2)) q2 FOC to obtain best response for follower 2 / q2 = ‐2q2 + (16 ‐ 2(q1 + q2)) = 0 q2 = BR2(q1) = (8 – q1) / 2 slides_414_15_16_2011_no_solutions 19 What is the optimal quantity in the first subgame, i.e. what is q1? In SPNE 2 responds w/ BR2(q1) and 1 knows so will use this information 1 = (16 ‐ 2(q1 + q2)) q1 = (16 – 2(q1 + ((8 – q1) / 2)) q1 = (16 – 3 q1 + 8) q1 FOC: 1 / q1 = – 3 q1 + (16 – 3 q1 + 8) = 0 → q1 = 4 Stackelberg equilibrium payoffs and first mover advantage … slides_414_15_16_2011_no_solutions 20 OTHER APPLICATIONS: TIMING & 1ST VS. 2ND MOVER ADVANTAGE Sometimes there is 1st mover advantage First established brand in a new product market where product becomes synonymous with brand Network effects and switching costs. When the size of a firm's network is important to its profitability, the firm that moves first might accumulate a sufficiently comprehensive network to deter later entry. Other times there is no 1st mover advantage Economics: Developing new products when there is considerable uncertainty can confer a large advantage to 2nd mover. Golder and Tellis (1993) find that in 36 new product markets the pioneer failed 47% and had market shares substantially lower than early market leaders (who were not pioneers) Electoral competition: An incumbent politician may already have staked out positions during his term of office. A challenger can then observe this and locate herself in policy space so as to gain the most votes. Sports: In the NHL, the home team is allowed to make the last substitution during all timeouts. slides_414_15_16_2011_no_solutions 21 Many strategic situations involve a trade‐off between acting sooner vs. later. These are timing games where the strategy involves a decision about when to take an action and fall under two categories Preemption games: Definition: those where the payoff is higher when (a) acting before others (so there is a 1st mover advantage) but (b) earlier actions are costlier (conditional on the order of who acts first). Outcomes: tend to lead players to move “too early” Example: open seating in planes or movie theater may lead all to arrive “too early” to secure a good seat. Attrition games: Definition: those where the payoff is higher when (a) acting after others (so there is no 1st mover advantage) but (b) later actions are costlier (conditional on the order of who acts first). Outcomes: tend to lead players move “too late” Examples: protracted trench wars w/out “decisive” action, delayed economic reforms slides_414_15_16_2011_no_solutions 22 Simple war of attrition example Two firms (or countries) are engaged in competition (war) where one gains 100 if the other exits (surrenders) first and immediately. The immediate exiter gets 0 Each period of inaction costs both some amount (e.g. 10) (indicating cost of waiting, e.g. resources to keep company alive or soldiers active) If after four potential rounds nobody acted assume they randomly decide who exits by tossing a coin so the expected value is 10=(1/2)*(100)+ (1/2)*(0) ‐ 10*4 slides_414_15_16_2011_no_solutions 23 Exercise: What is the SPNE? Is there a way for the players to achieve a higher expected payoff? slides_414_15_16_2011_no_solutions 24 DO PEOPLE REASON USING BACKWARD INDUCTION? Is there any Experimental Evidence for Backward Induction? Exercise: Consider the Centipede game played in class and derive the SPNE Solution: According to backward induction Player 1 should grab all on the first round At last round 2 faces two alternatives: grab 12.8 or get 6.4 in final round: so grab If 2 acts as above then at penultimate round 1 can grab 6.4 or get 3.2 in next round …until 1st round where 1 can grab .4 or get .2 next period slides_414_15_16_2011_no_solutions 25 Class evidence First treatment 13/15 first players stopped in 1st round 13/15 second players stopped 2nd round (their first decision) average stopped on first round but 2 went to last round slides_414_15_16_2011_no_solutions 26 Last treatment 4/14 first players stopped first round 4/14 second players stopped 2nd round (their first decision) Farthest: only 1 pair as far as 5th round average went to third round Winner Ekaterina: $8.80 Evidence from other classes: less support for prediction than our class (e.g. only 1% of first players grab initially) slides_414_15_16_2011_no_solutions 27 Does the experimental evidence using centipede game imply that people do not reason using backward induction? Not necessarily When the final stage gets close we see a large fraction of grabs, so perhaps people only reason backwards a limited amount of steps Perhaps the payoffs are incorrect and should take into account not just the money but also non‐pecuniary costs arising from grabbing if there is altruism If you are selfish but think the other may be altruistic then you may want to leave and let it grow as you expect the other not to grab it until close to the end. slides_414_15_16_2011_no_solutions 28 A Logical Paradox with Backward Induction: is it consistent with common knowledge of rationality? Consider centipede game: if we find ourselves at 5th round then why would player 1 grab? B/c otherwise expect 2 to grab in next round if 2 is rational But if 2 is rational and believes that 1 is also then why didn’t 2 (and 1) grab in an earlier round? How did we ever get to that round? Either the payoffs are incorrect or rationality is not common knowledge or there is a small probability of mistakes: players really are rational and know it but make small mistakes (hit continue key instead of stop) slides_414_15_16_2011_no_solutions 29 ...
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