dynamic games

This version: 8. March 2021 University of St. Gallen Dennis Gärtner Course Nr.: 4,200 | 4,202 Microeconomics III Supplementary Notes for: 2.2 Dynamic Games, Complete Information An example: Gibbons’ grenade game (4). This example is taken from Gibbons’ book, pp. 55-56 and is meant to illustrate one of the central issues in dynamic games: noncredible threats. Bear in mind that the example stems from a time in which it was perhaps more common for terrorists etc. to not want to kill themselves, hence the payoff of −∞ for both when the grenade is blown. I’ve elected to keep it because this reminds of the fact that we are still dealing with games of complete information, meaning that every player knows every other player’s preferences over outcomes. Specifically, in the example at hand, the assumption is that player 1 knows that player 2’s least preferred outcome is to blow the grenade (with or without money). By the way, if, like me, you find the game a bit violent – think of the grenade as a (perhaps rather nasty) stink bomb. Please note that the matrix (on slide 4) depicts only payoffs which result from possible outcomes of the described game (i.e., preferences over outcomes). It does not represent the game. Why? Because it does not convey the timing of moves described in the text. So we’ll have to find a different repre-sentation – which in fact is what we’ll do next. In case you’re following along in Gibbons’ book: notice that, after this motivating example, we’re in fact jumping to Section 2.4 of that book. (Gibbons has a somewhat unorthodox structure in chapter 2: he starts with a lot of examples of games and classes of games, saving the generic definitions for the chapter’s very last section.) Extensive form (6-7). On an intuitive level, it should be fairly clear what it means that we depict dy-namic games as a “multi-person decision tree”. It is worth pointing out though, that the notion of “tree” implies a bit more structure than might be obvious at first glance. For instance, the following is not a proper game tree: This example violates the property that “every terminal node has a unique history” (slide 7). And it doesn’t “branch out from its root” like a proper tree. Information sets and imperfect information (8-10). If the game on slide 9 strikes you as rather con-trived, then here’s a story to go with it: Imagine a committee-decision situation in which 2 players (your boss and her boss) must decide whether or not to approve your holiday request. First, player/su-perior 1 can say “yes” or “no”, then player/superior 2 can say “yes” or “no” (and after that, you as player 3 can either make them both a present or ignite a stink bomb). If the procedure is such that the holiday is granted only if both say “yes”, and if you as player 3 learn only the outcome (i.e. whether the holiday was granted or not) but not who said “yes” and who said “no”, then the game would have exactly the structure depicted in the example. Note: These notes were written in Spring 2020 to help make up for cancelled lectures. I am shar-ing them this year because you might find them useful. However, please note that they are not kept up to date, so some references (to slides, events, etc.) may well be outdated. 2 A remark: Nodes which are not connected to other nodes via info sets are sometimes interpreted as being contained in a singleton information set, i.e. an information set containing just one node. This can simplify certain formulations. Games with simultaneous moves can be represented in extensive form (11). This insight is quite use-ful for two reasons: First, it implies that extensive-form games are a generalization of normal-form games rather than being something different. Second (and perhaps more importantly), it makes clear that, from a strategic viewpoint, in the prisoners’ dilemma example, it doesn’t matter whether we literally think of our two prisoners being interrogated simultaneously, or in sequence, with the second not knowing what the first said. (This second point will become even clearer in a second, when we introduce the notion of strategies.) Definition of the extensive form (12-15). Try to internalize the (informal) definition (slide 12), but don’t spend an excessive amount of time on the ensuing formal definition (taken from the Mas-Colell-book). You certainly don’t need to know the latter by heart. It serves two purposes. First, for any of you who might think that all this drawing of trees etc. is not very concise and mathematical: Rest assured that it is absolutely possible to formulate a concise mathematical definition of a game in extensive form. Second, a useful takeaway for our purposes is the naming of nodes as a way to compactly refer to nodes when we need to (which we will in just a second, when we talk about strategies). The idea is: since any node is associated with a unique history of prior actions, let’s just name nodes by that history of prior actions (and the initial node will be the empty set). See the example on slide 15, where the set 𝑋𝑋 lists all nodes using this nomenclature (including terminal nodes 𝐸𝐸). An important note on exposition. As regards our motivating example (the grenade game), you are at this point in the slides in a position to formulate that game in extensive form (can you?). And having done that, you might be tempted to say: I know how to find the equilibrium – I’ll simply backward induct, as I’ve learned in previous courses! And you would not be wrong to do so. Nonetheless, our journey here will take a slightly different course. We will: 1) define what a strategy is; 2) define (actu-ally: recall) what a Nash equilibrium is; 3) observe that this is not an entirely satisfactory prediction; 4) refine this equilibrium using the notion of “subgame perfection”; 5) argue that subgame-perfect equilibrium produces the same equilibrium notion as backward induction (if correctly interpreted). Now you might wonder: why would we do all that, if we’ll wind up with backward induction anyhow? There are at least three reasons: • On a conceptual level, this enables us to relate our equilibrium concept for dynamic games to its counterpart for static games, in the sense that one is a “refinement” of the other. • On a more practical level, there are games in which it is not immediately clear what performing “backward induction” means, but where the notion of “subgame perfect equilibrium” is much more straightforward. We will see examples, namely games with multiple equilibria. • This more structured (but more tedious) approach will leave you better prepared for further topics such as games of incomplete information, or infinitely repeated games (which have no end, so there is in fact no end to backward induct from). Alternatively, you can trust me. 3 Strategies (16-20). It is crucial that you clearly understand and internalize the notion/definition of a strategy in dynamic games (as you might guess from the warning sign). It is not quite as trivial (and innocuous) as it may seem at first glance. To make sure that you have it down, go back to the examples on the previous slides, and ask yourself: how many possible (pure) strategies are there for any player in any of the examples?1 Example 1 Example 2 Example 3 Example 3 illustrates a potential pitfall when it comes to correctly applying the definition of “strategy”: player 1 has 4 available strategies – 2 nodes to decide, 2 available actions per node, so 2×2=4 strategies. One of these is {𝑅𝑅,(𝐿𝐿,𝐿𝐿′)→𝐿𝐿′′} (i.e., choose R at the initial node, and L’’ at the second). Which may seem odd at first, because: if player 1 chooses R at the first node, there’s no way he’ll ever get to decide at the other node – the game is sure not to proceed to that node. But: per definition of a strategy, an action must indeed be described for every node, so there are 4 possible strategies! And as you will see later, this understanding of the concept of strategy is in fact crucial for our ensuing equilibrium concepts to work, so bear with me. By the way, to efficiently figure out the number of available strategies, it helps to recall some very basic elementary combinatorics: suppose a player chooses at 𝐾𝐾 nodes 𝑘𝑘=1,…,𝐾𝐾 and that she has 𝑛𝑛𝑘𝑘 choices available at any node 𝑘𝑘. Then the number of possible (pure) strategies is 𝑛𝑛1×𝑛𝑛2×⋯×𝑛𝑛𝑘𝑘. If she has the same number of choices 𝑛𝑛 available at every node, this becomes 𝑛𝑛𝐾𝐾. For completeness, let me point out that it is perfectly possible to let players play mixed strategies also in extensive form games. To keep things as simple as possible, we will ignore that possibility for now and restrict players to playing pure strategies: it will play no role in the games we shall consider, and it actually raises some conceptual issues which we would need to further think about. A note on strategies in games of imperfect information: In games with non-trivial info sets, strategies can be understood (and notated) in two equivalent ways, namely 1. as a function which assigns an action to each info set, or 2. as a function which assigns an action to each node, with the additional restriction that nodes in the same info set are prescribed the same action. The formal definitions on the slides follow the second approach. Nonetheless, to keep notation com-pact, our following examples will frequently use the first. As an example: in the extensive-form prison-ers’ dilemma on slide 11, you could write prisoner 2’s strategy space as 𝑆𝑆2={𝑐𝑐,𝑑𝑑}, or as 𝑆𝑆2={{𝑐𝑐→𝑐𝑐,𝑑𝑑→𝑐𝑐},{𝑐𝑐→𝑑𝑑,𝑑𝑑→𝑑𝑑}}. The important thing to realize is that both represent the same set of strategies, it is only the notation that differs. 1 Example 1: P1 has 2, P3 has 2×2=4; Example 2: P1 has 2, P2 has 4, P3 has 24=16. Example 3: P2 has 2, P1 has 2×2=4. 4 Nash equilibria in the grenade game (22). This (identifying all Nash equilibria in a 2-player game) is a conceptually important step, and you should absolutely have a go at it yourself, as there is only so much you will learn by watching me do it. Proceed as follows: 1) determine each player’s full set of available strategies; 2) write down a table/matrix to show combinations of players’ strategies; 3) for each combination of strategies, figure out the eventual outcome (how does the game end if each player adheres to this strategy?) and note players’ payoffs from this outcome in the table; 4) having thus completed the table, find Nash equilibria in the usual fashion. [Note: if you failed to really try this on your own in the first round, you can always try this on other games, such as for instance the market-entry game on slide 31 (which, incidentally, is Rubinstein’s Example 162.1).] For supplementary read-ing: If you feel you need an alternative explanation of the procedure, check Tadelis’ Section 7.3 (in the context of a sequential Battle-of-the-Sexes game). Subgames (25-27). A different way to formulate the requirement for subgames in games with imper-fect info is: the boundary of the subgame (the red line on slide 25) is not permitted to intersect any player’s information set. Can you identify all subgames on slide 26?2 It may be worth bearing in mind that this definition will eventually (shortly, in fact) serve a purpose: subgames will identify parts of the game where we think it makes sense to check separately whether players are behaving (or would behave) rationally when/if we get to that part of the game. So, beyond all the formal definition, you should eventually develop a feel for a subgame as simply being a part of the game in which it is sensible to apply these additional checks. Subgame perfection in the grenade game (29). So the requirement of subgame perfection eliminates two of the three Nash equilibria in the grenade game. While it is common to say that “subgame per-fection eliminates non-credible threats”, the example makes clear that it in fact does more than that: it also eliminates the equilibrium in which P2 blows the grenade if and only if he does get the money. Which is indeed non-credible, but not much of a “threat” to P1. So more generally, subgame perfection eliminates on-credible strategies – whether interpretable as threats, or interpretable as “stupid” (i.e., with no strategic value, even if the player could commit). Example: a game of market entry (31-32). This is a classic example (from the field of “Industrial Or-ganization”) to illustrate the mechanics of non-credible threats. Notice that “fight” here is shorthand for the idea that a firm sets a lower price than it otherwise would, thereby hurting the other firm (but also its own) payoffs – this structure is reflected in the payoffs. (In case you’re interested in a more full-fledged, explicit model of this, check Example/Section 7.22 in Rubinstein’s book on page 213.) The variation in which both players pick an action upon entry is an example of an extensive-form game in which the smallest subgame is not a one-player game. Hence, backward induction requires us to first find the Nash equilibrium of that subgame. (If you’re looking for additional explanation: Example 208.3/207.1 in Rubinstein’s book, see p. 208, has the same structure, even if in the context of “Battle of the Sexes”.) Can you find Nash equilibria of this game which are not subgame perfect? (If you cannot find them intuitively, you can use the procedure which we used in the grenade game. With some prac-tice, you will quickly spot them without, though.) At this point, you could also play around with the game a bit and alter payoffs in the subgame after entry. Specifically: can you tweak payoffs so that there is/isn’t a non-subgame-perfect Nash equilibrium? Might you perhaps even be able to find a way to tweak payoffs so that there are two subgame-perfect Nash equilibria? This game is also an excellent opportunity to once more practice (and understand) our concept of a strategy. Specifically: is “out” a valid/complete strategy for player E? (Hint: No.) Why? The simple rea-son is: see our previous definition of a strategy. And the deeper reason is this: there’s no way to check, for instance, whether the profile of “strategies” (E:out; M:f) is a subgame-perfect equilibrium. But we 2 Answer: there are two, including the entire game itself (so just one “proper” subgame, beginning a player 3s rightmost node). 5 can check, for instance, whether (E:out;f;M:f) is a subgame-perfect equilibrium. This is why we defined strategies in the way we did (and why I was pedantic about it), because concepts only work with that notion of a strategy. The crucial implication for the learning process being: a solid understanding of concepts requires a clear and solid understanding of the foundation on which it builds. An example: Schoggistengeli-negotiations (34). This example illustrates the comment on the previous slide, that “[i]f a subgame has multiple Nash equilibria, then backward induction must be conducted ‘in all variations.’” Here, the subgame after P1 offered “(2,0)” has two equilibria: it is optimal for P2 to say “yes” or to say “no”.3 Consequently, either by conducting backward induction `in all variations’, or by first finding all Nash equilibria and then eliminating the ones which are not subgame perfect, you will find that the game has two subgame-perfect Nash equilibria.4 Notice how these are indeed both equilibria – there is no sense in which we can exclude one or the other. In particular, there is no sense in which P2 can “pick” the equilibrium in which he gets to keep both Schoggistengel: given that player 2 declines the offer (2,0), which is rational for player 2, it would not be rational for player 1 to offer anything other than (1,1). So this is indeed nothing other than the usual issue of having multiple equi-librium predictions, all of which are perfectly valid. It is one thing to formally establish these two equilibria (by arguing that neither can profitably deviate in any subgame, given the other's strategy), and you should absolutely make sure that you have a good grip on that. It is yet another to intuitively understand (or “rationalize”) the existence of these two equilibria. And, just in case you find yourself having trouble with that, this might be a good moment for a subtle but conceptually important point. We already discussed in our previous part 2.1 (on static games) how Nash equilibrium relies on more than just players’ rationality (and common knowledge thereof): it precludes players from having incorrect (even if rational) beliefs regarding other players’ choice of actions/strategies (or, put differently: it assumes players are able to “coordinate”). To the extent that we are still using Nash (even if refined), that feature (or bug – you be the judge) naturally extends to our present analysis of dynamic games. In particular, in the present Schoggistengel-exam-ple, rationality alone would not rule out that player 1 plays strategy (2,0) (because he/she thinks player 2 plays “yyy”, which is a perfectly rational assumption) and that player 2 plays strategy “nyy”, leading to a rejected offer and therefore equilibrium payoffs of (0,0)! This is much like rationality alone did not rule out that Chris and Pat go to different places. But it is not a subgame-perfect Nash equilibrium.5 A recipe: 2-stage games of perfect information (35). This describes the class of games which Gibbons’ book devotes Section 2.1 to. No, you don’t need to read that section, but it is worth noting that many interesting games fit this very simple structure – not least, the Stackelberg-game on the following slide. In terms of insights, it’s worth noting that this class of games is actually only a (small) extension of the grenade game, difference being that each player now has a richer set of actions (instead of the two which they had in the grenade game). So it is really just the grenade game, but with “more branches” for each player. When actions become continuous (and branches become very many), we of course start to have a hard time drawing things, so we need a shorthand. Two commonly used ways to indicate continuous choices are (on the right-hand side): 3 Don’t be confused by the notation for P1s actions in this example: actions are simply labelled by payoffs which they imply (if the offer is accepted). But they are still actions. 4 Which are: P1: (2,0), P2: yyy, and P1: (1,1), P2: nyy, where P2’s strategy is notated as how he responds to of-fers (2,0), (1,1) and (0,2), respectively. 5 Rationality alone similarly does not rule out player 1 playing strategy (1,1) and player 2 plays strategy “yyy”. However, that is perhaps a bit less striking because it leads to the same outcome as the SPNE-strategy-profile ((1,1),nyy). 6 Independently of the notation which you use: it is important not to lose sight of the fact that, despite the shorthand, player 2 has not just one but many (a continuum of!) nodes for which we must formu-late a decision. Which is why player 2’s strategy is described as a function, mapping player 1’s prior action into an action for player 2. Try not to lose sight of this relation between continuous-choice games and the prior discrete-choice examples. As regards the comment “provided that 𝐵𝐵2(𝑎𝑎1) is a function” in the proposed recipe: this is the con-tinuous-action pendant of saying “if each subgame has a unique Nash-equilibrium”. If not, we must proceed as in the Schoggistengel-example. Stackelberg duopoly (‘sequential Cournot’). This example of such a two-step game with continuous actions should be familiar (from Micro II). If you need more of a refresher than the slides offer, see Gibbons’ section 2.1.B (or just about any other book on Game Theory). Notice how it is easy to establish that the leader must be at least as well off in Stackelberg as she was in Cournot (i.e., in the simultaneous-choice version of the game). This is because, in Stackelberg, she could always choose the same quantity as in Cournot which, since the other firm plays best response in both cases, will lead to the same outcome and profits. It can also be argued (although not quite as easily) that the follower is worse off than in Cournot. Realizing that (as we discussed in the context of the Prisoners’ dilemma), the simultaneous-move ver-sion of the game (i.e. Cournot) can be understood as Stackelberg (i.e. sequential choice) but where the follower doesn’t observe the leaders quantity choice, this might seem puzzling at first: it means that the follower is worse off in the version of the game in which he knows more! The resolution to this apparent puzzle of course lies in the fact that the leader knows that the follower observes her choice, so it has strategic value for the leader. Despite its apparent simplicity, this is actually a valuable insight in a wide range of applied contexts! At the same time, it is also worth noting that it’s not always better to be the leader! Can you think of a counterexample?6 Dynamic bargaining games (38-44). The situation considered here is one in which one individual (the “seller” S) owns a good but values it at zero, whereas another player (the “buyer” B) has valuation 1 for the good. Clearly, there is a surplus of 1 from trade, but the question is how this surplus should be split (if trade takes place), as determined by the price 𝑝𝑝 at which the good trades hands. For supplementary reading, check Gibbons’ example in section 2.1.D (Gibbons directly starts with a version of the alternating-offers version with discounting, which is our “model 3”). For more extensive background reading, Osbourne and Tadelis both devote entire chapters to this class of problems, Os-bourne in Chapter 16 (“bargaining”, but don’t forget to look also at the Ultimatum game, or “model 1”, in Section 6.1.1), Tadelis in Chapter 11 (“strategic bargaining”). 6 If you can’t immediately think of one: how about taking a break and playing a round of sequential rock-paper-scissors (or matching pennies) against your buddy? 7 Before proceeding with the analysis, it is worth pointing out that this basic problem – how to divide surplus from trade – is a very fundamental one which pops up in all sorts of situations, be it when selling goods, when negotiating international trade agreements (between countries), in wage negoti-ations between firms and workers/unions, etc. Personally, to fix ideas, I like to present the analysis in the context of simply “splitting cake”, where the cake is the surplus of 1, and 𝑝𝑝 simply denotes the share of cake/surplus which player S gets.7 Returning to the model analysis: As regards the classical prediction (i.e., market equilibrium if both behave as price takers), the way I like to illustrate this is by drawing demand and supply curves for this “market”. Which will immediately show you that any price 𝑝𝑝 between 0 and 1 will equate supply and demand for the good. A technical remark: we assume here that the good which is up for trade is indivisible – it changes hands (from seller to buyer) or it doesn’t, but player cannot trade subdivisions (“parts”) of the good. Note that the classical prediction is not completely meaningless: it says that the good will change hands. But it says almost nothing about the price at which it will trade – and thereby, the division of surplus from trade. So we might hope to learn (and be able to predict) a bit more by more explicitly modeling the bargaining process using Game Theory. Model 1: the “ultimatum game” (39). The first game assumes that S makes an offer which B can either accept or reject, after which the game ends. In the literature, this has come to be called the “ultimatum game”. Notice that this game is an extension of the previous “Schoggistengel”-example, the extension being: in that game, there were only three (discrete) divisions of surplus available (and available sur-plus was two rather than one), whereas now, there are many possible divisions (a continuum, in fact). As regards (possibly non-subgame-perfect) Nash equilibria: it’s a good exercise for you to check the claim made on the slides, which is that for any 𝑝𝑝􀷤∈[0,1], there exists a Nash equilibrium such that trade takes place at that price 𝑝𝑝􀷤. As noted, you can achieve this using a strategy for the buyer of the form {‘yes’ iff 𝑝𝑝≤𝑝𝑝􀷤} (where “iff” is not a typo, but shorthand for: “if and only if”, meaning the buyer 7 In fact, for more accessible intuition, it is useful to think of players as directly proposing splits of surplus rather than a price. Technically, since there is a one-to-one mapping between price and split of surplus, this is equiva-lent. 8 declines all other offers). 8 Downside of course being: this prediction is no more useful than the classical prediction.9 But we haven’t checked for noncredible threats / subgame perfection yet… Doing that, we quickly see that the buyer must rationally accept all offers 𝑝𝑝<1 (anything else consti-tutes a non-credible threat), but that the subgame after which S offers 𝑝𝑝=1 has two equilibria: it is equally optimal for B to say yes or to say no. So, as in the Schoggistengeli-game, we must backward induct considering both possibilities. Considering the seller’s best response, however, we quickly see that there cannot exist a subgame-perfect Nash equilibrium in which B declines the offer 𝑝𝑝=1. The reason for this is the same which led us to conclude that Bertrand-competitors have no best response to the competitor setting a price 𝑝𝑝𝑗𝑗∈(𝑐𝑐,𝑝𝑝𝑚𝑚): If B rejects 𝑝𝑝=1 (but accepts any 𝑝𝑝<1), then S’s optimization problem has no maximum – for any 𝑝𝑝<1, S would be strictly better off by offering a slightly higher 𝑝𝑝. So S has no best response (to be precise: the set of best responses is empty). At the same time, notice how the continuity of actions makes the multiple equilibria from the discrete version disappear (see the Schoggistengeli-example above). So the only subgame-perfect equilibrium has B accepting all offers 𝑝𝑝≤1 (and reject all others), S of-fering 𝑝𝑝=1, implying that 1) trade takes place, and 2) S gets the entire surplus (or: cake). Which is certainly a more precise prediction than the classical one. But this comes at the cost of specific (and debatable) assumptions: that S (rather than B) gets to make the offer, and that the game ends (and surplus goes to waste!) if the other rejects – despite its obvious inefficiency. To see the importance of who gets to make the offer, I suggest you analyze the game with reversed roles, i.e. supposing that B offers a price 𝑝𝑝 (at which he is willing to buy), which S can accept or reject. You will see that, in this case, B gets the entire surplus/cake! By the way, the prediction of this model is sometimes contested based on the idea that, as B, you might decline low offers (high prices) because you would want to “punish” S, or “take revenge”, even if this comes at a cost to you (of foregoing your share, even if it is low). This is not a bad thought, and some experimental results (where less extreme splits are observed) seem to lend support to this idea. At the same time, I find it important to point out that this would be less of an instance of “our theory being wrong”, but our having to think about re-specifying preferences for such players, in that they care not only about their own wealth, but they are willing to forego own wealth in order to reduce the other’s wealth (for low levels of own wealth, at least). Such preferences might indeed lead to a different pre-diction.10 At the same time though, how realistic and how sensible such preferences are, is open for debate. Many professional business negotiators will advise you not to let feelings get in the way of your own material interests. At the same time, they might tell you that, in certain instances, it is not a bad idea to let your partner think that you are “crazy”, or: not materially rational. Which is ultimately the point in this game: as B, you will benefit if S thinks you are revengeful – this (and its strategic effect) is in a sense more crucial than your actually being revengeful. Which is a point we will return to in our final discussion, because it naturally leads over to models where other players’ preferences are up for debate, i.e. to models of incomplete information. 8 So what remains for you to do is to figure out: What is the seller’s best response to the proposed strategy? Can you check that the buyer’s strategy in turn constitutes a best response to that? 9 As a matter of fact, there are even more Nash equilibria than those pointed out so far, including equilibria in which no trade takes place, such as for instance: S proposes 𝑝𝑝=1, and B rejects any offer. In this sense, the Nash prediction is even weaker than the classical one, as it leaves a larger set of possible outcomes. 10 If this intrigues you, then you might for instance want to work out: What if B’s utility is 𝑢𝑢𝐵𝐵=𝑥𝑥𝐵𝐵−𝑥𝑥𝑆𝑆, where 𝑥𝑥𝑖𝑖 are players’ (direct) payoffs in the original game above? You will quickly work out that, with these prefer-ences, B would indeed only accept offers 𝑝𝑝≤1/2 (buying at price 𝑝𝑝 now gives B utility (1−𝑝𝑝)−𝑝𝑝=1−2𝑝𝑝, whereas no trade still gives utility 0−0=0). Can you think of other, perhaps more plausible specifications for preferences for “revenge”, or perhaps for some kind of “fairness”? Or for “envy”? 9 Model 2: alternating offers (40). In this variant, the game doesn’t end after B rejects, but now B gets to make an offer, which S can accept or reject. In the subgame starting with Bs offer 𝑝𝑝𝐵𝐵, you will check that the only subgame-perfect equilibrium has B offering 𝑝𝑝𝐵𝐵=0 (and S accepting any 𝑝𝑝𝐵𝐵≥0), so the unique SPNE has B get all the surplus in this subgame. Working further up the game tree, we see that this means that in the first round (when S makes the offer), B will reject any offer 𝑝𝑝𝑆𝑆>0 (because it gives him less than what he knows he will get in the subgame after he rejects), so in any subgame-perfect equilibrium, B must get the full surplus. Note that this game in fact has multiple subgame-perfect equilibria (as a nice exercise: can you describe them?), but they all share this property. As an exercise, can you work out what happens if we add yet another round, in which S again makes an offer? And what about yet another round (in which B makes an offer)? Etc. What you will quickly see is: whoever gets to make the final offer gets the entire surplus. Trivial as this might seem (after looking at it long enough), this is not useless advice for any real-life bargaining situation. Reinterpreting the insight a bit, it means: if you can, you want to influence the bargaining structure such that you get to make the final offer, i.e. such that the other party believes the game will end (and the cake will vanish) if they turn down your offer. Which, in terms of more specific examples, might remind you of current Brexit-negotiations between the EU and the UK and the (credible?) threats of walking away with no deal. Model 3: alternating offers with discounting (41). Returning to our cake-splitting problem, having es-tablished that the last one to make an offer gets the whole surplus, we might wonder whether this advantage could somehow counteracted by the cost of passing time, i.e., technically speaking, by “dis-counting”. Which in turn might be because of discounting (a preference for receiving any given share of surplus sooner rather than later), or because the cake indeed loses value over time.11 Whatever the reason, we assume here that if players cannot agree in the first round, the available surplus shrinks from 1 to 𝛿𝛿<1. The subgame starting with Bs offer is the same as before, only scaled down by factor 𝛿𝛿, meaning that B gets the full available surplus of 𝛿𝛿. Hence, B now rejects any offer 𝑝𝑝𝑆𝑆>1−𝛿𝛿. Which means that, in the unique subgame-perfect equilibrium, S offers 𝑝𝑝𝑆𝑆=1−𝛿𝛿 and B accepts, giving payoffs of 1−𝛿𝛿 to S and 𝛿𝛿 to B. Notice: trade takes place immediately (i.e. in the first round), which is efficient – once we introduce discounting, there is an efficiency cost to waiting! Second, even though B gets to make the final offer, S can now extract some of the surplus because waiting is costly in terms of reducing the size of the available cake. It is perhaps instructive to observe that this game nests the previous two models and predictions: for 𝛿𝛿=0 (the cake immediately goes bad after the first round), model and prediction correspond to model 1, whereas for 𝛿𝛿=1, we have model 2. And intermediate levels of 𝛿𝛿 will obvi-ously result in intermediate divisions of surplus, as the two effects interact. Model 4: repeated alternating offers (42-45). If you’re wondering how these two counteracting effects peter out in a model in which the game doesn’t abruptly end after the second round but can potentially continue for N rounds (if players keep rejecting): this is exactly what model 4 considers. The analysis is not out of reach at this point, it is absolutely doable (illustrating what fun things we could in principle do at this point, if we had more time), but it is cumbersome, which is why it is optional. As far as results and insights are concerned: It turns out that as we take the number of potential rounds N toward infinity, trade continues to take place in the very first round, and payoffs approach 1/(1+𝛿𝛿) and 𝛿𝛿/(1+𝛿𝛿) for S and B, respectively. Which in turn, as 𝛿𝛿→1, approaches one half each, i.e. an even split of surplus! Notice that 𝛿𝛿→1 might be interpreted as either 1) players not discounting and the cake not going bad, or 2) players discounting, but taking very, very quick turns with their alternating offers (so periods become very short). 11 Which makes this a bad model of sharing “Linzer Torte”, though – a cake known for getting better over time. 10 Playing against yourself: time consistency of preferences (45-49). This example considers a game in which you somewhat schizophrenically play against yourself (more precisely: against different in-stances in time of yourself). Which is a fun thing to do in itself, but perhaps more importantly, lays a valuable foundation. In many later courses, and in applications going far beyond microeconomics, you are likely to consider problems in which individuals (or firms, or governments…) must repeatedly make choices over time. It is commonly assumed (and compatible with a lot of empirical evidence), that in so doing, decisionmakers “discount the future”, which, in a nutshell, means: they strictly prefer having 𝑥𝑥 dollars today over having 𝑥𝑥 dollars in the future. This is typically captured by a “discount factor”, which “discounts” tomorrow’s utility when evaluated from today’s point of view. Moreover, when more than two periods are involved, you will usually assume exponential discounting, whereby dis-count factors for future periods form a geometric sequence, so starting with today, they are: 1,𝛿𝛿,𝛿𝛿2,𝛿𝛿3,… The game discussed here gives you an idea of why we usually assume this, by ways of illustrating the peculiarities and challenges when we drop the assumption, and people consequently no longer have time-consistent preferences. The mechanics of the application itself will be covered in the exercises. If you are nevertheless looking for supplementary reading, you may want to check Rabin’s original 1998 AER-article, mentioned in the slides – much of it is quite accessible, even to undergrads. Also, Tadelis’ textbook has a nice related discussion in Section 8.3.4 on “time-inconsistent preferences”. Technically, the bottom line of the exercise is to illustrate how non-exponential discounting gives rise to time-inconsistent behavior. To put the latter into context, it is perhaps worth pointing out that time-inconsistency in itself can be a troublesome assumption, in terms of an argument related to the “money pumps” which we saw in the context of uncertainty. There, we saw how individuals whose preferences do not satisfy the von-Neumann-Morgenstern axioms can be exploited as money pumps. A related argument can be made whenever individuals have time-inconsistent preferences. Here’s how:12 • Imagine that Alex has a lasting preference for $11 in 24 hours over $10 in 22 hours, but prefers $10 immediately over $11 in two hours. Note that such preferences are incompatible with geometric discounting, but compatible with hyperbolic discounting. • Now savvy Helen offers to sell him $11 in 24 hours for $10 in 22 hours. Alex agrees. 22 hours later, as Alex is about to give Helen the $10, his preferences have changed: he would now strictly prefer keeping the $10 over getting $11 in two hours. Thus, Helen offers to nullify the deal in exchange for some (small) amount such as, say, one cent. Given Alex’ (current) prefer-ences, he will be happy to accept – and Helen will be one cent richer. • If Helen keeps going in this fashion, she can extract arbitrary amounts from Alex, thus using Alex as a “money pump”, until Alex is ruined. Based on this insight, one may again make a Darwinian type of argument for why we might expect exponential discounters (who are invul-nerable to such money-pumps) to proliferate in society, and time-inconsistent hyperbolic dis-counters to go broke and vanish. A final remark: consistency of preferences is not to be confused with stationarity of preferences, i.e. there is nothing inconsistent in preferences changing over time. You might prefer meat to veggies today and you might prefer veggies to meat tomorrow, and there is nothing inconsistent about that. 12 This argument is adapted from: Loewe, G., 2006: “The development of a theory of rational intertemporal choice” Paper Rev. Soc. 80, 195–221, available at https://core.ac.uk/download/pdf/13270349.pdf. In case you’re interested, this paper also provides a great (and comprehensive) overall background reading on the topic of intertemporal choice. 11 Inconsistency comes into play, however, whenever you do not anticipate this happening (or, more generally, the chance of it happening). 13 Limits of subgame perfection (50-52). Intuitively appealing as it is, it may come as a bit of a surprise that the concept of “backward induction” (and subgame-perfect Nash equilibrium) is not as uncontest-able an equilibrium concept as it might seem at first. This is what the examples (or “paradoxes”) in this section are intended to illustrate. To put these caveats into perspective: it is worth noting beforehand that all these “paradoxes” will involve a player moving more than once. If you feel you require more background reading: unlike most of this chapter, these examples are not covered in Gibbons’ book, but you will find them discussed, for instance, in Osbourne’s Section 7.7. To appreciate the first example, it is important that you take your time and dwell on this game a little bit, putting yourself in players’ shoes. Eventually, you should see that, as player 2, you would perhaps not want to pick “d”, as prescribed by backward induction / subgame perfection. This is because you only get to choose (u or d) if player 1 deviated from playing what he should rationally play by backward induction – so how could you reasonably assume that he will return to being rational if you let him choose again? Notice how payoffs are deliberately chosen so that its only “a little bit” irrational to choose u instead of d at P1’s second node, but “very irrational” not to choose u at the very first node. This is meant to strengthen then idea that, as player 2, you would perhaps not want to rely on player 1’s rationality in the subgame. The second example is the “centipede game” – named after the predatory arthropod which its game tree resembles. It is a game between two players, and you can make sense of the structure of payoffs with a story like this: there is a “cake” of total surplus which grows over time (from 2 to 200, one unit at a time). While the cake grows, players alternate in having the opportunity to stop the game, in which the cake is evenly split, except: if player 2 stops the game he gets 1.5 units more than the even split (and player 1 gets 1.5 units less). As you will check, the only subgame perfect equilibrium has players stop in any round, leading to payoffs of 1 each. Which is of course grossly inefficient relative to the payoff of 100 each which they could have received, had both always continued. But that is not an equilibrium because it “unravels” from the end: player 2 can have a bit more by stopping in the round before. Anticipating this, player 1 in turn is better off stopping than waiting, etc. (As mentioned, for additional reading, this game is discussed in Osborne’s Section 7.7, but also in Tadelis’ Section 8.3.1, p. 159. Both also discuss interesting experimental lab results which show that players indeed don’t always play the subgame-perfect equilibrium.14) For the last example, the “chain store paradox”, it is important that you try to picture (not necessarily draw) the entire game tree and convince yourself that the only subgame-perfect Nash equilibrium has the monopolist accommodating every entrant (if it enters), and each entrant entering.15 The eventual point being that, by an argument related to that in the first example, if I were potential entrant #11, and if I saw the monopolist fight all previous 10 entrants (which he shouldn’t, in equilibrium!), then I’m not so sure I would want to bet my business on him suddenly reverting to playing according to that equilibrium prediction after I enter. 13 Two slightly more meaningful examples of changing preferences: some argue that children have a stronger taste for sugar than adults, and some argue that risk aversion increases with age. 14 Although, as Tadelis reports, grand masters in chess have been observed to always end the game immedi-ately! Which can be taken as evidence that it is decisive whether players think that the assumption of joint ra-tionality holds. 15 If you have trouble imagining the entire game tree, try drawing it, but I suggest setting 𝐾𝐾=2 or 𝐾𝐾=3 at most! 12 Which brings us to potential reactions to these paradoxes: 1. As touched upon already, we might assume that for some exogenous reasons, there are some-times random perturbations to players’ choices, i.e. that players become “temporarily insane” (or, less extremely put, they make mistakes). If this is what we believe in (and we deem these perturbations sufficiently unlikely), then we’re well advised to stick to subgame-perfect Nash analysis. 2. We might be tempted to think that our opponents are not just temporarily, but more funda-mentally insane. Problem being: it is hard to predict or advise much if we don’t think our op-ponents’ choices can somehow be rationalized. So a perhaps “softer” and more rationalizable form is: 3. We might think that other players are rational, but we may wonder whether we are correct in our assessment of other players’ preferences. Recall to this end that our analysis thus far as-sumes that players know each other’s preferences – this is what “complete information” means. In the game at hand, this means that as entrants, we are assumed to know that the monopolist prefers not to fight any of us. If we observe the monopolist (repeatedly) fighting, rather than calling him (temporarily) insane, we might start to wonder whether he has differ-ent preferences (i.e.: payoffs) than we thought. Which is a neat way to motivate our next chap-ter on games of incomplete information.16 16 The fun doesn’t end there, of course, as you will quickly realize that this introduces a strategic motive for the monopolist to pretend to have a preference for fighting even when he doesn’t – so as to deter entrants from entering the market. Which is a rationale which is at heart of many dynamic games of incomplete information.