Dynamic Games of complete information: Backward

3.2.2. Dynamic Games of complete information: Backward
Induction and Subgame perfection - Credibility and Commitment -

Strategic Behavior in Business and Econ
Outline 3.1. What is a Game ? The elements of a Game 3.1.2 The Rules of the Game: Example Examples of Game Situations 3.1.4 Types of Games 3.2. Solution Concepts Static Games of complete information: Dominant Strategies and Nash Equilibrium in pure and mixed strategies Dynamic Games of complete information: Backward Induction and Subgame perfection Strategic Behavior in Business and Econ

Reminder Dynamic games are games in which players move sequentially In dynamic games, the players should look forward (to understand the game) and think backwards (to solve the game) Dynamic games are better represented in tree form Backward Induction is the technique that finds only the “good” (sequentially rational) equilibria in games with Perfect Information Subgame Perfection is the technique that finds only the “good” (sequentially rational) equilibria in games with Imperfect Information Backward Induction and Subgame Perfection show that “incredible threats” (or “incredible promises”) should not be taken into account What a player knows about the choices of other players is as important as what the other players know about what you know about their choices Strategic Behavior in Business and Econ

Credibility and Commitment We have seen that “threats” or “promises” are not credible if they are not sequentially rational You don't become credible just by “talking” To become credible, commitment is necessary Strategic Behavior in Business and Econ

The market entry game A firm wants to enter a new market, and its main concern is about the reaction of an incumbent company that currently is making a profit $100,000 If the reaction is aggressive, the challenger will suffer a loss of $10,000, and the incumbent's profits will be reduced to only $20,000 (because of the costs of the fight being $80,000). On the other hand, if the incumbent chooses to accommodate to the new market scenario, then the two companies will share the $100,000 profit ($50,000 each) Obviously, the challenger can always choose to stay out if that seems to be the best choice (with a profit of $0) Strategic Behavior in Business and Econ

The game represented (in tree form) (- 10,000 , 20,000) Aggressive Firm 2 Enter Accommodate Firm 1 (50,000 , 50,000) Stay out (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in table form) Firm 2 Firm 1 Strategic Behavior in Business and Econ

The game represented (in table form) There are two Nash Equilibria Firm 2 Firm 1 Strategic Behavior in Business and Econ

The game represented (in table form) 1) Stay Out, Aggressive 2) Enter, Accommodate Firm 2 Firm 1 Strategic Behavior in Business and Econ

The game represented (in tree form) 1) Stay Out, Aggressive 2) Enter, Accommodate (- 10,000 , 20,000) Aggressive Firm 2 Enter Accommodate Firm 1 (50,000 , 50,000) Stay out (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) 1) Stay Out, Aggressive Is a Nash Equilibrium (- 10,000 , 20,000) Aggressive Firm 2 Enter Accommodate Firm 1 (50,000 , 50,000) Stay out (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) Given that Firm 1 is Staying Out, Firm 2 is fine with the Aggressive strategy (Its payoff is 100,000) In fact, if Firm 1 Stays Out, any strategy is a “Best Reply” for Firm 2 (- 10,000 , 20,000) Aggressive Firm 2 Enter Accommodate Firm 1 (50,000 , 50,000) Stay out (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) Given that Firm 2 is Aggressive, Firm 1 is fine with the Stay Out strategy (Its payoff is 0, which is better than -10,000) Given Firm 2's choice, Stay Out is a “Best Reply” for Firm 1 (- 10,000 , 20,000) Aggressive Firm 2 Enter Accommodate Firm 1 (50,000 , 50,000) Stay out (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) Aggressive is a Best Reply to Stay Out Stay Out is a Best Reply to Aggressive (- 10,000 , 20,000) Aggressive Firm 2 Enter Accommodate Firm 1 (50,000 , 50,000) Stay out (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) Firm 2 uses its strategy as a “Threat”. But the threat is incredible. If Firm 1 Enters, Firm 2 won't carry out its threat !! (- 10,000 , 20,000) Aggressive Firm 2 Enter Accommodate Firm 1 (50,000 , 50,000) Stay out (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) This equilibrium won't be selected by Backward Induction (- 10,000 , 20,000) Aggressive Firm 2 Enter Accommodate Firm 1 (50,000 , 50,000) Stay out (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) 2) Enter, Accommodate Is the other Nash Equilibrium (- 10,000 , 20,000) Aggressive Firm 2 Enter Accommodate Firm 1 (50,000 , 50,000) Stay out (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) Given that Firm 1 Enters, Firm 2 is better off by choosing Accommodate (a payoff of 50,000 instead of 20,000) Accommodate is a Best Reply (- 10,000 , 20,000) Aggressive Firm 2 Enter Accommodate Firm 1 (50,000 , 50,000) Stay out (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) Given that Firm 2 Accommodates, Firm 1 is better off by choosing Enter (a payoff of 50,000 instead of 0) Enter is a Best Reply (- 10,000 , 20,000) Aggressive Firm 2 Enter Accommodate Firm 1 (50,000 , 50,000) Stay out (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) Accommodate is a Best Reply to Enter Enter is a Best Reply to Accommodate (- 10,000 , 20,000) Aggressive Firm 2 Enter Accommodate Firm 1 (50,000 , 50,000) Stay out (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) Here there are not “incredible threats” This equilibrium is the one selected by Backward Induction (- 10,000 , 20,000) Aggressive Firm 2 Enter Accommodate Firm 1 (50,000 , 50,000) Stay out (0, 100,000) Strategic Behavior in Business and Econ

How to commit ? Consider the following modification: Before the game starts, Firm 2 makes an investment of $20, in an advertisement campaign. This investment will only be profitable in reducing the cost of being aggressive (from 80,000 to 60,000) If Firm 1 stays out, the profit will be 100,000 – 20,000 = 80,000 (the whole market profit minus the advertisement cost) If Firm 1 enters and Firm 2 Accommodates, the profit will be 50,000 – 20,000 = 30,000 (half the market profit minus the advertisement cost) If Firm 1 enters and Firm 2 is Aggressive, the profit will be 100,000 – 20, ,000 = 20,000 (the whole market profit minus the advertisement cost minus the fight cost) Strategic Behavior in Business and Econ

The game represented (in tree form) The game changes Notice !! The investment reduces (almost) ALL the payoffs for Firm 2 !!! (- 10,000 , 20,000) Aggressive (- 10,000 , 20,000) Firm 2 Enter Accommodate Firm 1 (50,000 , 30,000) (50,000 , 50,000) Stay out (0, 80,000) (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) What is NOW the outcome the game by Backward Induction ? (- 10,000 , 20,000) Aggressive (- 10,000 , 20,000) Firm 2 Enter Accommodate Firm 1 (50,000 , 30,000) (50,000 , 50,000) Stay out (0, 80,000) (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) Still the threat of being aggressive is incredible !! Nothing changes with respect to the previous situation (- 10,000 , 20,000) Aggressive (- 10,000 , 20,000) Firm 2 Enter Accommodate Firm 1 (50,000 , 30,000) (50,000 , 50,000) Stay out (0, 80,000) (0, 100,000) Strategic Behavior in Business and Econ

How to commit ? Consider the following modification instead: Before the game starts, Firm 2 makes an investment of $40, in an advertisement campaign. This investment will only be profitable in reducing the cost of being aggressive (from 80,000 to 45,000) If Firm 1 stays out, the profit will be 100,000 – 40,000 = 60,000 (the whole market profit minus the advertisement cost) If Firm 1 enters and Firm 2 Accommodates, the profit will be 50,000 – 40,000 = 10,000 (half the market profit minus the advertisement cost) If Firm 1 enters and Firm 2 is Aggressive, the profit will be 100,000 – 40, ,000 = 15,000 (the whole market profit minus the advertisement cost minus the fight cost) Strategic Behavior in Business and Econ

The game represented (in tree form) The game changes Notice !! The investment reduces ALL the payoffs for Firm 2 !!! (- 10,000 , 15,000) Aggressive (- 10,000 , 20,000) Firm 2 Enter Accommodate Firm 1 (50,000 , 10,000) (50,000 , 50,000) Stay out (0, 60,000) (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) What is NOW the outcome the game by Backward Induction ? (- 10,000 , 15,000) Aggressive (- 10,000 , 20,000) Firm 2 Enter Accommodate Firm 1 (50,000 , 10,000) (50,000 , 50,000) Stay out (0, 60,000) (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) Now the threat is credible !! By making the right investment Firm 2 shows that it's committed to the market (- 10,000 , 15,000) Aggressive (- 10,000 , 20,000) Firm 2 Enter Accommodate Firm 1 (50,000 , 10,000) (50,000 , 50,000) Stay out (0, 60,000) (0, 100,000) Strategic Behavior in Business and Econ

The game represented (in tree form) Although Firm 2 is reducing ALL its payoffs, at the end makes more money (60,000) than in the previous case (50,000) (- 10,000 , 15,000) Aggressive (- 10,000 , 20,000) Firm 2 Enter Accommodate Firm 1 (50,000 , 10,000) (50,000 , 50,000) Stay out (0, 60,000) (0, 100,000) Strategic Behavior in Business and Econ

Summary Incredible threats won't be taken into account (bluffing) If you want to be credible, you must commit “Hurt yourself to help yourself” Reduce your payoffs, or your strategies, to show you are not bluffing If done right, this is enough to convince your opponents that your threats are credible But, once again, it is important that your opponents know that your are committed ! This is like “burning bridges” or “throwing the steering wheel out of the window”, it must be done with care and in the right way Strategic Behavior in Business and Econ

Dynamic Games of complete information: Backward Induction and Subgame perfection - Some Games - Strategic Behavior in Business and Econ

Some Games Military recruitment during the Civil War Cold War (revisited) Financial Crisis (“Toy” version) Strategic Behavior in Business and Econ

Military recruitment during the Civil War ( ) During the Civil War, recruitment was an important issue for the two sides. According to the law, if you were called to serve you could not refuse. The only way to avoid being called is that someone else is called. (Recruitment was fully random) There was also an option to become volunteer. Being a volunteer was rewarded Being called to served had a high cost (higher than the reward) The “dilemma” was: Should I wait to see if I am not called or should I become a volunteer to get a reward ? Strategic Behavior in Business and Econ

Military recruitment during the Civil War ( ) The Game Let us assume, for simplicity, that there are only two players in town Player 1 chooses first and must decide between being Volunteer or Wait Then Player 2, knowing the choice by Player 1, must decide also between being Volunteer or Wait If one of the players becomes volunteer, the game is over Otherwise, one of the players is called to serve “at random” (50% chances) The “reward” for being volunteer is of $300. No “reward” if you are called The “cost” of serving is of $400 (either for being volunteer or for being called to serve) If not serving, the payoff is 0 (no reward, no cost) Strategic Behavior in Business and Econ

Military recruitment during the Civil War ( ) The Game represented (tree form) -100, 0 Volunteer Player 1 0, -100 Volunteer Wait Player 1 is called -400, 0 Player 2 (½) Wait Player 0 (½) Player 2 is called 0, -400 Strategic Behavior in Business and Econ

Military recruitment during the Civil War ( ) Expected Payoffs (for each player) if both wait E(Wait,Wait)=( ½ )·(-400) + ( ½ )·(0) = -200 Player 1 Wait Player 1 is called -400, 0 Player 2 (½) Wait Player 0 (½) Player 2 is called 0, -400 Strategic Behavior in Business and Econ

Military recruitment during the Civil War ( ) The Game represented (tree form) -100, 0 Volunteer Player 1 0, -100 Volunteer Wait Player 2 Wait -200, -200 Strategic Behavior in Business and Econ

Military recruitment during the Civil War ( ) By Backward Induction -100, 0 Volunteer Player 1 0, -100 Volunteer Wait Player 2 Wait -200, -200 Strategic Behavior in Business and Econ

Military recruitment during the Civil War ( ) By Backward Induction – Final Outcome -100, 0 This would be the final outcome of the game Player 1 Waits while Player 2 goes Volunteer Volunteer Player 1 0, -100 Volunteer Wait Player 2 Wait -200, -200 Strategic Behavior in Business and Econ

Military recruitment during the Civil War ( ) By Backward Induction – Final Outcome -100, 0 Player 2 prefers the sure loss of -100 rather than the risk of being recruited Volunteer Player 1 0, -100 Volunteer Wait Player 2 Wait -200, -200 Strategic Behavior in Business and Econ

Military recruitment during the Civil War ( ) By Backward Induction – Final Outcome -100, 0 Knowing that, Player 1 prefers to Wait Volunteer Player 1 0, -100 Volunteer Wait Player 2 Wait -200, -200 Strategic Behavior in Business and Econ

Military recruitment during the Civil War ( ) By Backward Induction – Final Outcome -100, 0 Player 2 is a “Reluctant Volunteer” Volunteer Player 1 0, -100 Volunteer Wait Player 2 Wait -200, -200 Strategic Behavior in Business and Econ

Military recruitment during the Civil War ( ) By Backward Induction – Final Outcome -100, 0 Notice that Player 2 might threat Player 1 with Waiting And then, Player 1 would choose to Volunteer Volunteer Player 1 0, -100 Volunteer Wait Player 2 Wait -200, -200 Strategic Behavior in Business and Econ

Military recruitment during the Civil War ( ) By Backward Induction – Final Outcome -100, 0 That would be a (Nash) equilibrium. But once again, it involves an incredible threat. It won't be carried out Volunteer Player 1 0, -100 Volunteer Wait Player 2 Wait -200, -200 Strategic Behavior in Business and Econ

Military recruitment during the Civil War ( ) By Backward Induction – Final Outcome -100, 0 This is the unique equilibrium that is credible It exhibits a “first mover advantage” Volunteer Player 1 0, -100 Volunteer Wait Player 2 Wait -200, -200 Strategic Behavior in Business and Econ

Military recruitment during the Civil War ( ) 2.1 million soldiers served in the Civil War Only 50,000 were recruited 96 % of the soldiers were volunteers !! (reluctant ?) Strategic Behavior in Business and Econ

Cold War (revisited) (based on “Game theory and the Cuban missile crisis” by Steven J. Brams) (based on “Mutually Assured Destruction” by Roy Gardner) This is a real life (historical) version of a Game of Chicken. It goes back to the 60's, when the USA and the USSR were engaged in a so called “cold war” One of the episodes of the “cold war” was the Cuban missile crisis The Cuban missile crisis was precipitated by a Soviet attempt in October 1962 to install medium-range and intermediate-range nuclear-armed ballistic missiles in Cuba that were capable of hitting a large portion of the United States. The goal of the United States was immediate removal of the Soviet missiles, and U.S. policy makers seriously considered two strategies to achieve this end Strategic Behavior in Business and Econ

A naval blockade (B), or "quarantine" as it was euphemistically called, to prevent shipment of more missiles, possibly followed by stronger action to induce the Soviet Union to withdraw the missiles already installed. A "surgical" air strike (A) to wipe out the missiles already installed, insofar as possible, perhaps followed by an invasion of the island. The alternatives open to Soviet policy makers were: Withdrawal (W) of their missiles. Maintenance (M) of their missiles. NEW !! If both countries escalate, a FINAL DECISION must be taken Strategic Behavior in Business and Econ

The environment of the game Players: USA and USSR Strategies: For USA: Blockade (B) or Air Strike (A) For USSR: Withdrawal (W) or Maintenance (M) (plus a FINAL DECISION) Payoffs: (to be defined) The Rules of the Game Timing of moves Simultaneous AND Sequential Nature of conflict and interaction Conflict (anti-coordination) Information conditions Symmetric Strategic Behavior in Business and Econ

The payoffs This is “Game of Chicken”. Both would like the other to “swerve”. If they “stay straight”, the crash is the Nuclear War USA USSR Best outcome (A,W) (B,M) 4 Second Best (B,W) (B,W) 3 Third Best (B,M) (A,W) 2 Worst Outcome (A,M) (A,M) 1 Strategic Behavior in Business and Econ

3, 3 2 , 4 USA wins 4 , 2 1 , 1 Cuban Missile crisis (static version)
We have two solutions in the static version of the Game USSR W M USSR wins 2 , 4 Compromise 3, 3 B USA USA wins 4 , 2 Nuclear War 1 , 1 A Strategic Behavior in Business and Econ

3, 3 2 , 4 USA wins 4 , 2 1 , 1 Cuban Missile crisis (static version)
Let us make the game “sequential” and defer the final decision of “Nuclear War” USSR W M USSR wins 2 , 4 Compromise 3, 3 B USA USA wins 4 , 2 Nuclear War 1 , 1 A Strategic Behavior in Business and Econ

Cuban Missile crisis (dynamic version) 3, 3 Withdraw USSR Blockade Maintain 2, 4 USA 4, 2 Withdraw Air Strike USSR USSR Maintain The Final Decision game USA Strategic Behavior in Business and Econ

Cuban Missile crisis (dynamic version) 3, 3 Withdraw What would be the outcome of the Final Decision Game ? USSR Blockade Maintain 2, 4 USA 4, 2 Withdraw Air Strike USSR USSR Maintain The Final Decision game USA Strategic Behavior in Business and Econ

Cuban Missile crisis (dynamic version) 3, 3 Withdraw The Final Decision Game has TWO equilibria 1) Doomsday,Doomsday 2) Retreat, Retreat USSR Blockade Maintain 2, 4 USA 4, 2 Withdraw Air Strike USSR USSR Maintain The Final Decision game USA Strategic Behavior in Business and Econ

Cuban Missile crisis (dynamic version) 3, 3 Withdraw Equilibrium of the whole game when the outcome of the Final Decision (sub)game is Doomsday, Doomsday USSR Blockade Maintain 2, 4 USA 4, 2 Withdraw Air Strike USSR Maintain 1 , 1 Outcome of the Final Decision game Strategic Behavior in Business and Econ

Cuban Missile crisis (dynamic version) 3, 3 Withdraw Equilibrium of the whole game when the outcome of the Final Decision (sub)game is Retreat, Retreat USSR Blockade Maintain 2, 4 USA 4, 2 Withdraw Air Strike USSR Maintain 3 , 3 Outcome of the Final Decision game Strategic Behavior in Business and Econ

In the two equilibria, the USA strategy at the beginning is the Same: Air Strike (that is, being aggressive) Then, in the first equilibrium the USSR Maintains the missiles while in the second Withdraws In some sense the United States "won" by getting the Soviets to withdraw their missiles (but Premier Nikita Khrushchev of the Soviet Union at the same time extracted from President Kennedy a promise not to invade Cuba, which seems to indicate that the eventual outcome was a compromise of sorts) According to game theory, if the USSR withdrew (first equilibrium) is because for some reason thought that the Doomsday threat was credible !! And according to Game Theory, it is credible indeed !! Strategic Behavior in Business and Econ

“To continue to deter in an era of strategic nuclear equivalence, it is necessary to have nuclear (as well as conventional) forces such that in considering aggression against our interests any adversary would recognize that no plausible outcome would represent a victory or any plausible definition of victory. To this end and so as to preserve the possibility of bargaining effectively to terminate the war on acceptable terms that are as favorable as practical, if deterrence fails initially, we must be capable of fighting successfully so that the adversary would not achieve his war aims and would suffer costs that are unacceptable, or in any event greater than his gains, from having initiated an attack.” President Carter in 1980, Presidential Directive 59, Nuclear Weapons Employment Policy Strategic Behavior in Business and Econ

Financial Crisis (Toy version) When we bring our money to a Bank we hope that the Bank is going to invest wisely We expect to have our money back plus some interest earnings But we do not know what the Bank does with our money (or, perhaps, we find out too late) Strategic Behavior in Business and Econ

Financial Crisis (Toy version) The Game An individual (the Costumer) has $100,000 to invest and has two alternatives: To buy a Treasury Bond (T-Bond) that has a 3% interest rate (for sure) To put the money in the Bank that offers a Certificate of Deposit (CD) that promises an interest rate of 10% (not insured) If the Bank gets the deposit it will invest the money in some financial derivative. Financial derivatives are very risky, but the risk can be reduced if the Bank puts a lot of effort in the search of the right investment. A good investment yields a 12% interest whereas a bad investment yields a -100% interest (total loss !) High effort: A good investment is found with a probability of 99% Low effort: A good investment is found with a probability of 90% The effort has a cost: $300 for the high effort, $100 for the low effort Strategic Behavior in Business and Econ

Financial Crisis (Toy version) The Game (payoff computations) For the Costumer From the T-Bond: 100, % = 103,000 From the CD: If good Investment: 100, % = 110,00 If bad Investment: 100, % = 0 For the Bank If the Costumer buys the T-Bond: 0 If the Costumer buys the CD: If good Investment with High effort: 100,000 * 2% = 1,700 If good Investment with Low effort: 100,000 * 2% = 1,900 If bad Investment with High effort: 100,000 * (- 100%) = -300 If bad Investment with Low effort: 100,000 * (- 100%) = -100 Strategic Behavior in Business and Econ

Financial Crisis (Toy version) The Game represented (tree form) 110,000 , 1700 Good investment Chance (0.99) High effort (0.01) Bad Investment Bank 0 , -300 110,000 , 1900 CD Good investment Low effort (0.90) Costumer Chance (0.10) T-Bond Bad Investment 0 , -100 103,000 , 0 Strategic Behavior in Business and Econ

Financial Crisis (Toy version) The Game (expected payoffs for the Bank) E(High effort) = (0.99)·(1,700) + (0.01)·(-300) = 1,680 E(Low effort) = (0.90)·(1,900) + (0.10)·(-100) = 1,700 The Game (expected payoffs for the Costumer) E(High effort) = (0.99)·(110,000) + (0.01)·(0) = 108,900 E(Low effort) = (0.90)·(110,000) + (0.10)·(0) = 99,000 Strategic Behavior in Business and Econ

Financial Crisis (Toy version) The Game represented (simplified tree form) 108,900 , 1,680 High effort Bank CD Low effort Costumer 99,000 , 1,700 T-Bond 103,000 , 0 Strategic Behavior in Business and Econ

Financial Crisis (Toy version) The Game represented (Backward Induction) 108,900 , 1,680 High effort Bank CD Low effort Costumer 99,000 , 1,700 T-Bond 103,000 , 0 The solution by Backward Induction shows that the Bank will Choose a low effort and then the Costumer will prefer the T-Bond (sure option) Strategic Behavior in Business and Econ

Financial Crisis (Toy version) The Game represented (Backward Induction) 108,900 , 1,680 High effort Bank CD Low effort Costumer 99,000 , 1,700 T-Bond 103,000 , 0 High effort would be an “incredible promise” Strategic Behavior in Business and Econ

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