DynamicGames

,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Dynamic Games,1,Overview,In this unit we study:,Combinations of sequential and simultaneous games,Solutions to these types of games,Repeated games,How to use dynamics to build self-sustaining agreements,2,Sequential and Simultaneous Games,There are many situations where the strategic situation has both simultaneous and sequential elements,Examples:,Decision by a firm to enter a market followed by competition in pricing and advertising,Decisions by candidates to run for office followed by voting,Attempts at legal settlements followed by trial in the event no settlement is reached,3,How to Analyze these Games?,In sequential games, we saw that it paid to look forward and reason back,Find the best decision a player can make on reaching a point in the game,“Prune” the game tree to eliminate worse (dominated) decisions,In simultaneous games, we looked for a best response to a best response (Nash equilibrium).,Project the strategy of a rival,Choose a best response to that strategy,Check if the rival would want to change his “projected” strategy in view of your move.,4,New Elements to Dynamic Games,History matters,Strategy is now based not only on projections of the future and the present but also on the past.,The,process,by which you arrived at a point in the game might matter,Alternative futures matter,Histories of the game that no one contemplated as arising can play a key role in influencing outcomes,5,Entry,Consider the following situation:,An incumbent firm is presently operating in some profitable market,A potential rival firm is considering entering this market,Upon entry, the rival and the incumbent simultaneously decide whether to fight or not fight,If they both fight, then the profitability of the market is such that the rival would be better off staying out,If they do not fight, the rival would be better off entering,6,Game Tree,rival,2a, e,out,in,Incumbent is row player.,Payoffs are (incumbent, rival),7,Description,One interpretation of this diagram is the following:,There are three levels of profitability in the market:,High (when no one fights),Medium (when 1 firm fights),Low (when both fight),The product of fighting is to capture market share from the rival,If both fight or both dont fight, market share is 50-50,If one fights and the other doesnt the fighter gains market share at the expense of the rival,8,Goal,Your goal is to provide an analysis of under what conditions to enter this market.,What are the key things to think about in making this decision?,9,Example 1: Large Market Share Capture,Suppose that the profitability of the market is:,Big (No one fights)= 32,Medium (One firm fights)= 25,Small (Both fight) = 16,The outside option of a rival who does not enter is 11,When 1 firm fights and the other does not, the fighter obtains 80% market share,10,Game Tree: Example 1,rival,32, 11,out,in,11,Analysis,If the rival enters:,Incumbents best response to dont fight is to fight,Incumbents best response to fight is to fight,The situation is symmetric for the rival,Therefore:,If enter, then (fight, fight),12,Game Tree: Example 1,rival,32, 11,out,in,Since the rival anticipates a fight on entering, it is better not to enter,13,Game Tree: Example 1- Generalized,rival,2a, e,out,in,a e b (It only pays to enter absent a fight),c a,b d,14,Game Tree: Example 1- Generalized,rival,2a, e,out,in,a e b (It only pays to enter absent a fight),c a,b d,Notice that the game after entry is a Prisoners dilemma,In this case the incumbent uses the b, b outcome to successfully deter entry,15,Comments,Notice that what,didnt happen,entry had a profound effect on what did,The rival could count on the fact that entry combined with the temptation to grab market share would lead to a fight,Therefore, it paid to stay out of this market.,16,Example 2: Smaller Market Share Grab,Suppose that the market sizes are again,Big = 32,Medium = 25,Small = 16,The outside option of a rival who does not enter is still 11,When 1 firm fights and the other does not, the fighter obtains,60%,market share,Now what happens?,17,Game Tree: Example 2,rival,32, 11,out,in,18,Analysis,If rival enters:,Incumbents best response to dont fight is dont fight,Incumbents best response to fight is dont fight,Same for rival,So neither fight if rival enters,If rival does not enter,Incumbent is free to threaten to do whatever it likes,In particular, it can threaten to fight,In which case it pays for the rival to stay out,19,Equilibria,Rival enters, neither firm fights,Rival doesnt enter, incumbent threatens to fight if it did enter,Notice that now entry deterrence depends crucially on the rivals,beliefs,about the incumbents response,If the rival is convinced that the incumbent will be aggressive, it should not enter,Since the rival chooses not to enter, choosing to actually be aggressive is a,best response,by incumbent,20,Game Tree: Example 2 Generalized,rival,2a, e,out,in,a e b,1. a c,2. d b,21,Game Tree: Example 2 Generalized,rival,2a, e,out,in,a e b,1. a c,2. d b,Even though it is a dominant strategy for incumbent to not fight,It can deter entry by threatening.,Since in the even of successful deterrence, the threat is no tested this is still a best response for the incumbent,22,Comments,So it would seem that if the incumbent can affect the rivals beliefs, it is possible to deter entry even in this framework.,23,Choosing Equilibria,The prospect that a threat which is costly to carry out might succeed in a situation like this posed a problem for game theory,Is there some rational means to choose between the equilibria?,24,Subgame Perfect Equilibria,Well generalize the idea of look forward, reason back the following way:,Rationality Axiom 2: When presented with any history of the game (even an unexpected one), players should choose best responses to future beliefs,Formally, we require that players choose optimizing strategies everywhere in the game,25,Example 2: Refined,Recall that not fighting was a dominant strategy for each of the players if the rival enters,Therefore, despite incumbents threats to the contrary,Rival should anticipate that its entry will not lead to fighting,Therefore, it pays to enter.,Entry deterrence is not,credible,in this case.,26,Example 3: Shrinking Markets,Now consider a variation of example 2.,Suppose that when either firm fights, it still gains 60% market share but the profitability of the market shrinks to a greater extent than before.,Does this change rivals view of the incumbents threat to fight?,27,Example 3: Specifics,Suppose that the market sizes are now,Big = 32,Medium =,18,Small = 16,The outside option of a rival who does not enter is 11,When 1 firm fights and the other does not, the fighter still obtains 60% market share,28,Game Tree: Example 3,rival,32, 11,out,in,29,Analysis,The best response to not fighting is not to fight,The best response to fighting is to fight,Therefore, there are 2 equilibria following entry,If rival anticipates a fight, it should not enter,If it anticipates no fighting, it should,Hence there are 2 equilibria of the dynamic game:,Out - Fight, Fight,In - Dont fight, Dont fight,30,Comments,Notice that making competition more disruptive on entry actually,improves,the credibility of the threat by the incumbent,In many situations it is possible to control how destructive competition will be in markets,The subtlety here is that its in the incumbents interest to make the destructiveness of competition more rather than less,31,Game Tree: Example 3 Generalized,rival,2a, e,out,in,a e b,1. a c,2. b d,32,Game Tree: Example 2 Generalized,rival,2a, e,out,in,a e b,1. a c,2. b d,Now this is a coordination game if the rival enters.,Either both firms can coordinate on not fighting, or they can coordinate on fighting,Either is self-sustaining and the fighting outcome deters entry.,33,Key Conclusions,The idea of subgame perfection is to assess the credibility of threats.,Well return to this issue in the next class,Threats which are not self-sustaining if carried out, should correctly be viewed with skepticism,To make a threat credible, it can sometimes serve the interest of the incumbent to,destroy,profitability of the market in the event of entry.,34,Repeated Games,We turn now to repeated games,These are games where players are involved in the same (or similar) strategic situation for many periods in a row.,The key insight here will be that we can use the,future,to affect the outcome in the present.,35,Example 1 Revisited,Suppose that entry has occurred and that the situation is as in example 1,36,Example 1: Game Table,Suppose,a b,Competition is destructive to profitability,c a,Market share grabs are profitable,b d,Fighting back is better than being a victim of a grab,37,Analysis,We determined that fighting was the inevitable outcome,With consequent decrease in the profitability of the market,Suppose that the firms will compete for 2 periods instead of 1?,Firm 1 and 2 agree to the following:,Dont fight in either period,If either of us fights in the first period, fight in the second,Will this work?,38,Carrots and Sticks,Each firm is holding out a carrotthe promise of a in both periods,And a stick, the threat of b in the second period,To try to deter the temptation to grab market share (and get c),This could work if 2a c + b,39,Look Forward Reason Back,But theres a problem here:,In period 2, theres no stick and no carrot,So each firm will be tempted to fight and succumb to that temptation,And this “reverberates” back to period 1,Each firm knows that there is no “carrot” in the second period-only the “stick”,So theres nothing to deter the temptation to fight in period 1,Hence the firms will fight in both periods,40,Generalizing,The principle applies for any set number of periods.,Since theres nothing to promote good behavior at the end of the game, firms will fight then,And this reverberates backward throughout the game,The conclusion is a sad one:,If the game has any set ending time, the firms will fight in every period,41,Infinite Repetition,Suppose there is no fixed endpoint to the game,Instead the firms expect the game to be,infinitely repeated,This is an abstraction-think of it as a game being repeated for a really long time with no one knowing exactly when it will end,Now can the firms cooperate?,42,Tit-for-Tat Strategies,Suppose the firms make the following deal:,We agree not to fight,If my rival fights, Ill fight in the next period,Then the war is over and well resume not fighting,Will this work?,Now in every period the firm must weigh the gains from cheating (i.e. the temptation),c a,Versus the cost in the future (the carrot and stick),a b,If the cost exceeds the temptation, each firm will refrain from fighting.,a b c a,43,Variations,Suppose the carrot and stick in the above agreement are not enough to overcome the temptation,i.e., a b c a,45,Key Point,The larger is the temptation or the weaker the punishment available in any one period,The longer the threat of “war” needs to be to deter cheating.,The size of the stick needs to be calibrated to the upside from cheating.,Obviously the promise of infinite war leads to the largest possible “stick”,46,Discounting,All of this assumed that profits in the future were worth the same as those in the present,Of course, theyre not,Suppose that we,discount,profits in the future by the real interest rate,r,.,How does this change the analysis?,47,Tit for,n,Tats,If we play tit-for-tat:,NPV of temptation = c a,NPV of the threat = (a b)/(1 +,r,),Notice that as the interest rate increases, the punishment associated with the threat declines,48,Perpetual Punishment,In a world with positive real interest rates, even the threat of perpetual punishment may not be enough to stave off fighting,Recall the perpetuity formula:,What is the NPV of an asset has a cash flow,C,starting 1 period in the future and lasting in perpetuity?,NPV =,C/r,49,Analysis,Suppose we threaten punishment forever after a defection.,NPV of Temptation = c a,NPV of punishment = (a b)/,r,For higher real interest rates, even the most severe possible punishment loses efficacy,If this threat is insufficient to deter cheating, nothing will deter it.,50,More Variations to Consider,It is enough that there is a possibility that the game will continue into the future to use threat to sustain cooperation,Like the interest rate, the higher the chance the game will end, the less powerful the threat and the harder to sustain agreement,Like the entry games, threat need to be credible-a strong punishment strategy that is not credible to implement is not an effective deterrent.,51,Using Threats to Sustain Cooperation,What makes for a good threat?,Detection,Clarity,Repetition and reputation,Credibility of enforcement,What about forgiveness?,What about errors?,52,Conclusion,This suggests:,Severe but credible punishments,Limited temptation from short run abuses,Strong upside incentives to cooperate.,If costs are increasing in the severity of each measure then a mix of measures is best.,53,