Category Archives: Bargaining

The Economics of John Nash

I’m in the midst of a four week string of conferences and travel, and terribly backed up with posts on some great new papers, but I can’t let the tragic passing today of John Nash go by without comment. When nonacademics ask what I do, I often say that I work in a branch of applied math called game theory; if you say you are economist, the man on the street expects you to know when unemployment will go down, or which stocks they should buy, or whether monetary expansion will lead to inflation, questions which the applied theorist has little ability to answer in a satisfying way. But then, if you mention game theory, without question the most common way your interlocutor knows the field is via Russell Crowe’s John Nash character in A Beautiful Mind, so surely, and rightfully, no game theorist has greater popular name recognition.

Now Nash’s contributions to economics are very small, though enormously influential. He was a pure mathematician who took only one course in economics in his studies; more on this fortuitous course shortly. The contributions are simple to state: Nash founded the theory of non-cooperative games, and he instigated an important, though ultimately unsuccessful, literature on bargaining. Nash essentially only has two short papers on each topic, each of which is easy to follow for a modern reader, so I will generally discuss some background on the work rather than the well-known results directly.

First, non-cooperative games. Robert Leonard has a very interesting intellectual history of the early days of game theory, the formal study of strategic interaction, which begins well before Nash. Many like to cite von Neumann’s “Zur Theorie der Gesellschaftsspiele” (“A Theory of Parlor Games”), from whence we have the minimax theorem, but Emile Borel in the early 1920’s, and Ernst Zermelo with his eponymous theorem a decade earlier, surely form relevant prehistory as well. These earlier attempts, including von Neumann’s book with Morganstern, did not allow general investigation of what we now call noncooperative games, or strategic situations where players do not attempt to collude. The most famous situation of this type is the Prisoner’s Dilemma, a simple example, yet a shocking one: competing agents, be they individuals, firms or countries, may (in a sense) rationally find themselves taking actions which both parties think is worse than some alternative. Given the U.S. government interest in how a joint nuclear world with the Soviets would play out, analyzing situations of that type was not simply a “Gesellschaftsspiele” in the late 1940s; Nash himself was funded by the Atomic Energy Commission, and RAND, site of a huge amount of important early game theory research, was linked to the military.

Nash’s insight was, in retrospect, very simple. Consider a soccer penalty kick, where the only options are to kick left and right for the shooter, and to simultaneously dive left or right for the goalie. Now at first glance, it seems like there can be no equilibrium: if the shooter will kick left, then the goalie will jump to that side, in which case the shooter would prefer to shoot right, in which case the goalie would prefer to switch as well, and so on. In real life, then, what do we expect to happen? Well, surely we expect that the shooter will sometimes shoot left and sometimes right, and likewise the goalie will mix which way she dives. That is, instead of two strategies for each player, we have a continuum of mixed strategies, where a mixed strategy is simply a probability distribution over the strategies “Left, Right”. This idea of mixed strategies “convexifies” the strategy space so that we can use fixed point strategies to guarantee that an equilibrium exists in every finite-strategy noncooperative game under expected utility (Kakutani’s Fixed Point in the initial one-page paper in PNAS which Nash wrote his very first year of graduate school, and Brouwer’s Fixed Point in the Annals of Math article which more rigorously lays out Nash’s noncooperative theory). Because of Nash, we are able to analyze essentially whatever nonstrategic situation we want under what seems to be a reasonable solution concept (I optimize given my beliefs about what others will do, and my beliefs are in the end correct). More importantly, the fixed point theorems Nash used to generate his equilibria are now so broadly applied that no respectable economist should now get a PhD without understanding how they work.

(A quick aside: it is quite interesting to me that game theory, as opposed to Walrasian/Marshallian economics, does not derive from physics or other natural sciences, but rather from a program at the intersection of formal logic and mathematics, primarily in Germany, primarily in the early 20th century. I still have a mind to write a proper paper on this intellectual history at some point, but there is a very real sense in which economics post-Samuelson, von Neumann and Nash forms a rather continuous methodology with earlier social science in the sense of qualitative deduction, whereas it is our sister social sciences which, for a variety of reasons, go on writing papers without the powerful tools of modern logic and the mathematics which followed Hilbert. When Nash makes claims about the existence of equilibria due to Brouwer, the mathematics is merely the structure holding up and extending ideas concerning the interaction of agents in noncooperative systems that would have been totally familiar to earlier generations of economists who simply didn’t have access to tools like the fixed point theorems, in the same way that Samuelson and Houthakker’s ideas on utility are no great break from earlier work aside from their explicit incorporation of deduction on the basis of relational logic, a tool unknown to economists in the 19th century. That is, I claim the mathematization of economics in the mid 20th century represents no major methodological break, nor an attempt to ape the natural sciences. Back to Nash’s work in the direct sense.)

Nash only applies his theory to one game: a simplified version of poker due to his advisor called Kuhn Poker. It turned out that the noncooperative solution was not immediately applicable, at least to the types of applied situations where it is now commonplace, without a handful of modifications. In my read of the intellectual history, noncooperative games was a bit of a failure outside the realm of pure math in its first 25 years because we still needed Harsanyi’s purification theorem and Bayesian equilibria to understand what exactly was going on with mixed strategies, Reinhard Selten’s idea of subgame perfection to reasonably analyze games with multiple stages, and the idea of mechanism design of Gibbard, Vickers, Myerson, Maskin, and Satterthwaite (among others) to make it possible to discuss how institutions affect outcomes which are determined in equilibrium. It is not simply economists that Nash influenced; among many others, his work directly leads to the evolutionary games of Maynard Smith and Price in biology and linguistics, the upper and lower values of his 1953 results have been used to prove other mathematical results and to discuss what is meant as truth in philosophy, and Nash is widespread in the analysis of voting behavior in political science and international relations.

The bargaining solution is a trickier legacy. Recall Nash’s sole economics course, which he took as an undergraduate. In that course, he wrote a term paper, eventually to appear in Econometrica, where he attempted to axiomatize what will happen when two parties bargain over some outcome. The idea is simple. Whatever the bargaining outcome is, we want it to satisfy a handful of reasonable assumptions. First, since ordinal utility is invariant to affine transformations of a utility function, the bargaining outcome should not be affected by these types of transformations: only ordinal preferences should matter. Second, the outcome should be Pareto optimal: the players would have to mighty spiteful to throw away part of the pie rather than give it to at least one of them. Third, given their utility functions players should be treated symmetrically. Fourth (and a bit controversially, as we will see), Nash insisted on Independence of Irrelevant Alternatives, meaning that if f(T) is the set of “fair bargains” when T is the set of all potential bargains, then if the potential set of bargains is smaller yet still contains f(T), say S strictly contained by T where f(T) is in S, then f(T) must remain the barganing outcome. It turns out that under these assumptions, there is a unique outcome which maximizes (u(x)-u(d))*(v(x)-v(d)), where u and v are each player’s utility functions, x is the vector of payoffs under the eventual bargain, and d the “status-quo” payoff if no bargain is made. This is natural in many cases. For instance, if two identical agents are splitting a dollar, then 50-50 is the only Nash outcome. Uniqueness is not at all obvious: recall the Edgeworth box and you will see that individual rationality and Pareto optimality alone leave many potential equilibria. Nash’s result is elegant and surprising, and it is no surprise that Nash’s grad school recommendation letter famously was only one sentence long: “This man is a genius.”

One problem with Nash bargaining, however. Nash was famously bipolar in real life, but there is an analogous bipolar feel to the idea of Nash equilibrium and the idea of Nash bargaining: where exactly are threats in Nash’s bargain theory? That is, Nash bargaining as an idea completely follows from the cooperative theory of von Neumann and Morganstern. Consider two identical agents splitting a dollar once more. Imagine that one of the agents already has 30 cents, so that only 70 of the cents are actually in the middle of the table. The Nash solution is that the person who starts with the thirty cents eventually winds up with 65 cents, and the other person with 35. But play this out in your head.

Player 1: “I, already having the 30 cents, should get half of what remains. It is only fair, and if you don’t give me 65 I will walk away from this table and we will each get nothing more.”

Player 2: “What does that have to do with it? The fair outcome is 50 cents each, which leaves you with more than your originally thirty, so you can take your threat and go jump off a bridge.”

That is, 50/50 might be a reasonable solution here, right? This might make even more sense if we take a more concrete example: bargaining over wages. Imagine the prevailing wage for CEOs in your industry is $250,000. Two identical CEOs will generate $500,000 in value for the firm if hired. CEO Candidate One has no other job offer. CEO Candidate Two has an offer from a job with similar prestige and benefits, paying $175,000. Surely we can’t believe that the second CEO will wind up with higher pay, right? It is a completely noncredible threat to take the $175,000 offer, hence it shouldn’t affect the bargaining outcome. A pet peeve of mine is that many applied economists are still using Nash bargaining – often in the context of the labor market! – despite this well-known problem.

Nash was quite aware of this, as can be seen by his 1953 Econometrica, where he attempts to give a noncooperative bargaining game that reaches the earlier axiomatic outcome. Indeed, this paper inspired an enormous research agenda called the Nash Program devoted to finding noncooperative games which generate well-known or reasonable-sounding cooperative solution outcomes. In some sense, the idea of “implementation” in mechanism design, where we investigate whether there exists a game which can generate socially or coalitionally preferred outcomes noncooperatively, can be thought of as a successful modern branch of the Nash program. Nash’s ’53 noncooperative game simply involves adding a bit of noise into the set of possible outcomes. Consider splitting a dollar again. Let a third party tell each player to name how many cents they want. If the joint requests are feasible, then the dollar is split (with any remainder thrown away), else each player gets nothing. Clearly every split of the dollar on the Pareto frontier is a Nash equilibrium, as is each player requesting the full dollar and getting nothing. However, if there is a tiny bit of noise about whether there is exactly one dollar, or .99 cents, or 1.01 cents, etc., when deciding whether to ask for more money, I will have to weigh the higher payoff if the joint demand is feasible against the payoff zero if my increased demand makes the split impossible and hence neither of us earn anything. In a rough sense, Nash shows that as the distribution of noise becomes degenerate around the true bargaining frontier, players will demand exactly their Nash bargaining outcome. Of course it is interesting that there exists some bargaining game that generates the Nash solution, and the idea that we should study noncooperative games which implement cooperate solution concepts is without a doubt seminal, but this particular game seems very strange to me, as I don’t understand what the source of the noise is, why it becomes degenerate, etc.

On the shoulders of Nash, however, bargaining progressed a huge amount. Three papers in particular are worth your time, although hopefully you have seen these before: Kalai and Smorodinsky 1975 who retaining the axiomatic approach but drop IIA, Rubinstein’s famous 1982 Econometrica on noncooperative bargaining with alternative offers, and Binmore, Rubinstein and Wolinsky on implementation of bargaining solutions which deals with the idea of threats as I did above.

You can read all four Nash papers in the original literally during your lunch hour; this seems to me a worthy way to tip your cap toward a man who literally helped make modern economics possible.

“The Efficiency of Dynamic Post-Auction Bargaining: Evidence from Wholesale Used-Auto Auctions,” B. Larsen (2013)

Another job market season is in the books, as Brad Larsen was the last job talk I made it to this year. Here’s something you may not have known: there is an industry which sells lots of used cars at auctions. Each year, they sell eighty billion dollars worth of cars at these auctions! I’ve written auction theory papers before, and yet have never even heard of this industry. The complexity of economics never fails to impress, but this is good; it means there is always something interesting for us economists to study.

Now when you hear “inefficient bargaining” and “autos”, you probably immediately think of Stiglitz’ lemons model. But selection really isn’t a big issue in these used-auto auctions. Car rental agencies that participate, for instance, sell off an entire season of cars all at once. The real inefficiency is of the Myerson-Satterthwaite type. In that classic result, your private value for a good and my private value for a good are drawn from overlapping distributions – and “draw” and “distribution” may be over the set of all possible buyers or sellers as in Harsanyi purification – then there is no price at which we would agree to a transaction. (You must know Myerson-Satterthwaite! Among other things, it absolutely destroys the insight behind the Coase theorem…) Myerson and Satterthwaite, as well as a handful of other others, later fully described the second-best mechanism when first-best trade is impossible. This second-best mechanism turns out to be quite complicated. Almost all of the auto auction houses use a seemingly strange mechanism instead. First, sellers set a secret reserve which is told to the auctioneer. Then the car is sold in an English auction. If the auction ends below the reserve price, then sellers and potential buyers can opt to bargain individually over the phone, perhaps asynchronously. At the end of these bargains, there are still many cars that go unsold.

This brings up the obvious question: how close to the second-best optimum is this strange selling mechanism? Larsen got access to some amazing data, which included outcomes of the auction, all of the sequential offers made over the phone, seller secret reserve prices, and tons of data at the level of the car and the buyer. The problem, of course, is that we don’t have access to the seller and buyer types (i.e., their private valuations of the car). Indeed, we don’t even have knowledge of the distribution of their types. If you are a regular reader of this blog, you know what is coming next – we can perhaps back out those distributions, but only if theory is applied in a clever way. (And, wow, how are job market speakers not permanently nervous? It’s one thing to stretch the Myerson-Satterthwaite framework to its limits, but it’s another thing to do so when Mark is sitting in the front row of the seminar room, as was the case here at Kellogg. Luckily, he’s a nice guy!)

The high bid in the auction helps us get buyer valuations. This high bid is, if bidders bid truthfully, equivalent to the second-order statistic of bidder values. Order theory turns out to give us a precise relationship between the underlying distribution and distribution of draws of its second-order statistic (when the second order is from a set of N draws from the distribution). Larsen knows exactly how many people were at each auction house, and can use this to roughly estimate how many bidders could potentially have bid on each car. Homogenizing each car using observable characteristics, therefore, the order statistic method can be used to gather the underlying distribution of buyer values for each car.

On the seller side, it is the secret reserve price that really helps us estimate the bargaining game. If the post-auction bargaining never results in a price lower than the last bid, then bidders have a dominant strategy to bid truthfully. Costly bargaining in addition to truthful bidding by buyers means that the optimal seller reserve price must be strictly increasing in the seller’s type. And what does strict monotonicity give us? Invertibility. We now know there is a link between the seller reserve price and the seller type. The structure of post-auction bargaining helps us to bound the relationship between seller type and the reserve. Finally, multiple rounds of bargaining let us bound the disutility costs of bargaining, which turn out to be quite small.

So how does the auto auction bargaining mechanism do? In this market, given the estimated type distributions, the Myerson-Satterthwaite second-best gives you 98% of the surplus of a first-best (informationally impossible) mechanism. Real-world dynamic bargaining captures 88-96% of that second-best surplus, with the most inefficiency coming from small sellers where, perhaps, screening problems result. That’s still leaving money on the table, though. Larsen, among others, suggests that these types of strange dynamic mechanisms may persist because in two-sided markets, they roughly are equivalent to committing to giving buyers a large chunk of the gains from trade, and we know that attracting additional potential buyers is the most important part of successful auctions.

January 2013 working paper (No IDEAS version).

“Long Cheap Talk,” R. Aumann & S. Hart (2003)

I wonder if Crawford and Sobel knew just what they were starting when they wrote their canonical cheap talk paper – it is incredible how much more we know about the value of cheap communication even when agents are biased. Most importantly, it is not true that bias or self-interest means we must always require people to “put skin in the game” or perform some costly action in order to prove the true state of their private information. A colleague passed along this paper by Aumann and Hart which addresses a question that has long bedeviled students of repeated games: why don’t they end right away? (And fair notice: we once had a full office shrine, complete with votive candles, to Aumann, he of the antediluvian beard and two volume tome, so you could say we’re fans!)

Take a really simple cheap talk game, where only one agent has any useful information. Row knows what game we are playing, and Column only knows the probability distribution of such games. In the absence of conflict (say, where there are two symmetric games, each of which has one Pareto optimal equilibrium), Row first tells Column that which game is the true one, this is credible, and so Column plays the Pareto optimal action. In other cases, we know from Crawford-Sobel logic that partial revelation may be useful even when there are conflicts of interest: Row tells Column with some probability what the true game is. We can also create new equilibria by using talk to reach “compromise”. Take a Battle of the Sexes, with LL payoff (6,2), RR (2,6) and LR=RL=(0,0). The equilibria of the simultaneous game without cheap talk are LL,RR, or randomize 3/4 on your preferred location and 1/4 of the opponent’s preferred location. But a new equilibria is possible if we can use talk to create a public randomization device. We both write down 1 or 2 on a piece of paper, then show the papers to each other. If the sum is even, we both go LL. If the sum is odd, we both then go RR. This gives ex-ante payoff (4,4), which is not an equilibrium payoff without the cheap talk.

So how do multiple rounds help us? They allow us to combine these motives for cheap talk. Take an extended Battle of the Sexes, with a third action A available to Column. LL still pays off (6,2), RR still (2,6) and LR=RL=(0,0). RA or LA pays off (3,0). Before we begin play, we may be playing extended Battle of the Sexes, or we may be playing a game Last Option that pays off 0 to both players unless Column plays A, in which case both players get 4; both games are equally probable ex-ante, and only Row learns which game we actually in. Here, we can enforce a payoff of (4,4) if, when the game is actually extended Battle of the Sexes, we randomize between L and R as in the previous paragraph, but if the game is Last Option, Column always plays A. But the order in which we publicly randomize and reveal information matters! If we first randomize, then reveal which game we are playing, then whenever the public randomization causes us to play RR (giving row player a payoff of 2 in Battle of the Sexes), Row will afterwards have the incentive to claim we are actually playing Last Resort. But if Row first reveals which game we are playing, and then we randomize if we are playing extended Battle of the Sexes, we indeed enforce ex-ante expected payoff (4,4).

Aumann and Hart show precisely what can be achieved with arbitrarily long strings of cheap talk, using a clever geometric proof which is far too complex to even summarize here. But a nice example of how really long cheap talk of this fashion can be used is in a paper by Krishna and Morgan called The Art of Conversation. Take a standard Crawford-Sobel model. The true state of the world is drawn uniformly from [0,1]. I know the true state, and get utility which is maximized when you take action on [0,1] as close as possible to the true state of the world plus .1. Your utility is maximized when you take action as close as possible to the true state of the world. With this “bias”, there is a partially informative one-shot cheap talk equilibrium: I tell you whether we are in [0,.3] or [.3,1] and you in turn take action either .15 or .65. How might we do better with a string of cheap talk? Try the following: first I tell you whether we are in [0,.2] or [.2,1]. If I say we are in the low interval, you take action .1. If I say we are in the high interval, we perform a public randomization which ends the game (with you taking action .6) with probability 4/9 and continues the game with probability 5/9; for example, to publicly randomize we might both shout out numbers between 1 and 9, and if the difference is 4 or less, we continue. If we continue, I tell you whether we are in [.2,.4] or [.4,1]. If I say [.2,.4], you take action .3, else you take action .7. It is easy to calculate that both players are better off ex-ante that in the one-shot cheap talk game. The probabilities 4/9 and 5/9 were chosen so as to make each player indifferent from following the proposed equilibrium after the randomization or not.

The usefulness of the lotteries interspersed with the partial revelation are to let the sender credibly reveal more information. If there were no lottery, but instead we always continued with probability 1, look at what happens when the true state of nature is .19. The sender knows he can say in the first revelation that, actually, we are on [.2,1], then in the second revelation that, actually, we are on [.2,4], in which case the receiver plays .3 (which is almost exactly sender’s ideal point .29). Hence without the lotteries, the sender has an incentive to lie at the first revelation stage. That is, cheap talk can serve to give us jointly controlled lotteries in between successive revelation of information, and in so doing, improve our payoffs.

Final published Econometrica 2003 copy (IDEAS). Sequential cheap talk has had many interesting uses. I particularly enjoyed this 2008 AER by Alonso, Dessein and Matouschek. The gist is the following: it is often thought that the tradeoff between decentralized firms and centralized firms is more local control in exchange for more difficult coordination. But think hard about what information will be transmitted by regional managers who only care about their own division’s profits. As coordination becomes more important, the optimal strategy in my division is more closely linked to the optimal decision in other divisions. Hence I, the regional manager, have a greater incentive to freely share information with other regional managers than in the situation where coordination is less important. You may prefer centralized decision-making when cooperation is least important because this is when individual managers are least likely to freely share useful information with each other.

“The Nash Bargaining Solution in Economic Modeling,” K. Binmore, A. Rubinsten & A. Wolinsky (1986)

If we form a joint venture, our two firms will jointly earn a profit of N dollars. If our two countries agree to this costly treaty, total world welfare will increase by the equivalent of N dollars. How should we split the profit in the joint venture case, or the costs in the case of the treaty? There are two main ways of thinking about this problem: the static bargaining approach developed first by John Nash, and bargaining outcomes that form the perfect outcome of a strategic game, for which Rubinstein (1982) really opened the field.

The Nash solution says the following. Let us have some pie of size 1 to divide. Let each of us have a threat point, S1 and S2. Then if certain axioms are followed (symmetry, invariance to unimportant transformations of the utility function, Pareto optimality and something called the IIA condition), the bargain is the one that maximizes (u1(p)-u1(S1))*(u2(1-p)-u2(S2)), where p is the share of the pie of size 1 that accrues to player 1. So if we both have linear utility, player 1 can leave and collect .3, and player 2 can leave and collect 0, but a total of 1 is earned by our joint venture, the Nash bargaining solution is the p that maximizes (p-.3)*(1-p-0); that is, p=.65. This is pretty intuitive: 1-.3-0=.7 of surplus is generated by the joint venture, and we each get our outside option plus half of that surplus.

The static outcome is not very compelling, however, as Tom Schelling long ago pointed out. In particular, the outside option looks like a noncredible threat: If player 2 refused to offer player 1 more than .31, then Player 1 would accept given his outside option is only .3. That is, in a one-shot bargaining game, any p between .3 and 1 looks like an equilibrium. It is also not totally clear how we should interpret the utility functions u1 and u2, and the threat points S1 and S2.

Rubinstein bargaining began to fix this. Let players make offers back and forth, and let there be a time period D between each offer. If no agreement is reached after T periods, we both get our outside options. Under some pretty compelling axioms, there is a unique perfect equilibrium whereby player 1 gets p* if he makes the first offer, and p** if player 2 makes the first offer. Roughly, if the time between offers is D, player 1 must offer player 2 a high enough share that player 2 is indifferent between that share today and the amount he could earn when he makes an offer in the next period. Note that the outside options do not come into play unless, say, player 1’s outside option is higher than min{p*,p**}. Note also that as D goes to 0, all of the difference in bargaining power has to do with who is more patient. Binmore et al modify this game so that, instead of discounting the future, rather there is a small chance that the gains from negotiation will disappear (“breakdown”) in between every period; for instance, we may want to form a joint venture to invent some product, but while we negotiate, another firm may swoop in and invent it. It turns out that this model, with von Neumann-Morganstern utility functions for each player (though perhaps differing levels of risk aversion) is a special case of Rubinstein bargaining.

Binmore et al prove that as D goes to zero, both strategic cases above have unique perfect equilibria equal to a Nash bargaining solution. But a Nash solution for what utility functions and threat points? The Rubinstein game limits to Nash bargaining where the difference in utilities has to do with time preference, and the threat points S1 and S2 are equal to zero. The breakdown game limits to Nash bargaining where the difference in utilities has to do with risk aversion, and the threat points S1 and S2 are equal to whatever utility we would get from the world after breakdown.

Two important points: first, it was well known that a concave transformation of a utility function leads to a worse outcome in Nash bargaining for that player. But we know from the previous paragraph that this concave transformation is equivalent to a more impatient Rubinstein bargainer: a concave transformation of the utilities in the Nash outcome has to do with changing the patience, not the risk aversion, of players. Second, Schelling was right when he argued that the Nash threat points involve noncredible threats. As long as players prefer their Rubinstein equilibrium outcome to their outside option, the outside option does not matter for the bargaining outcome. Take the example above where one player could leave the joint venture and still earn .3. The limit of Rubinstein bargaining is for each player to earn .5 from the joint venture, not .65 and .35. The fact that one player could leave the joint venture and still earn .3 is totally inconsequential to the negotiation, since the other player knows that this threat is not credible whenever the first player could earn at least .31 by staying. This point is often wildly misunderstood when people apply Nash bargaining solutions: properly defining the threat point matters!

Final RAND version (IDEAS). There has been substantial work since the 80s on the problem of bargaining, particularly in trying to construct models where delay is generated, since Rubinstein guarantees agreement immediately and real-world bargaining rarely ends in one step; unsurprisingly, these newer papers tend to rely on difficult manipulation of theorems using asymmetric information.


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