Res Politica versus Res Economica
September 18, 2011
Today’s Outside the Box is the latest chapter in my ongoing discussion with Dr. Woody Brock on the rationale of the politics of economics. In this essay, Woody explains how political science has taken a back seat to economics, and how to redress the imbalance we find today between what he terms “Res Politica” (the rule of politics) and “Res Economica” (the rule of economics or money). Where the rubber meets the road here is that our important economic decisions are increasingly being made by politicians (who are not particularly well-schooled in either economics or political science), with consequences that are likely to be dangerous. You will have to put on your thinking cap, but this will provide you with some real insights and food for thought.
Woody is one of the best “big-picture” economic theoreticians of our time, and that’s why I treasure the times we get to talk (or rather I get to “sit in “school” and learn), and have invited him to speak at our annual conference. He has already committed for next year, so save the dates: May 2-4 in La Jolla. In the meantime, you can find more of Woody’s thinking at his company’s site, Strategic Economic Decisions. (For the record, this is the first OTB I have sent from my iPad.)
Your hoping the politicians are listening analyst,
John Mauldin, Editor
Outside the Box
RES POLITICA versus RES ECONOMICA
Why Economics Must Yield to Politics as the Paradigm of Tomorrow
By: Horace W. Brock, Ph.D. President Strategic Economic Decisions, Inc., http://www.SEDinc.com
Author’s Note: We are living in the age of economists, the Age of Larry Summers as it were. But economists have less and less to say about the important issues of our time. This is because these issues are political – indeed political philosophical – in nature, not economic. Yet “political science” is rightly regarded as a second rate discipline, and political philosophy has morphed into the History of Political Thought.
This essay explains why political science became irrelevant, and how to redress today’s imbalance between Res Politica and Res Economica.
A. Economics Imperialism and its Origins
The phrase “economics imperialism” has circulated for nearly three decades. It refers to the reality that, of all the social sciences, economics has emerged as the most relevant, most useful, and most rigorous discipline. Its perspective on social behavior and its analytic methods have invaded every facet of sociology, political science, and social psychology. The success of such books as “Freakonomics” is proof of precisely this point, as has been trumpeted by its author Steven Levitt. Finally, if any further proof of economics hegemony is needed, just consider the surging enrollments in economics and finance courses at major universities worldwide, a surge that is well known to have caught university administrators off guard.
The same phenomenon is true in public policy analysis. There was a time when the cabinet of the US president was dominated by lawyers, or political scientists and theorists, but that is no longer the case. We are living in an age when economists such as Martin Feldstein or Lawrence Summers or Alan Greenspan dominate policy discussions. With their well-honed analytical skills (lacking in other fields), they sound off with credibility on any number of topics, and often have the last word.
There are four reasons why all this has happened. First, the discipline of economics is indeed highly analytical and rigorous, and this imparts credibility to it. It can explain phenomena, and also (to some extent) abet forecasting the future.
Second, the analytics of economics are not mere abstractions, but are transformed into testable models via the linkage between economics and econometrics. In an age when the “objectivity” of analysis is prized (and indeed required by the press), it sure helps a policy maker to trot out extensive statistical back-up for his case. The fact that most people confronting econometric evidence have no way of knowing whether the underlying statistical methodology is valid does not change this reality.
Third, economics was the first discipline to put central emphasis on the concept of “incentives.” When they make decisions, consumers, producers, and investors respond to given incentives. This point is extremely important for two reasons: (i) the concept of “incentive structure compatibility” is arguably the most important concept ever set forth in the history of analytical social science; and (ii) incentives can be changed by government policy. This second point has permitted economics to be linked to public policy in a very compelling manner: By knowing the consequences of changing incentives, a politician can much better predict the outcome of a change in policy, and thus identify a better policy.
Fourth, beginning students of economics are presented with a timeless and powerful analytical model that is as compelling to economics as is the Law of Gravity to physics: The Law of Supply and Demand. Imagine economics without this model! Moreover, no matter how far students progress in their studies, they never deviate far from the model of market equilibration via the price system.
The Contrasting Failure of Political Science: Now contrast this plethora of selling points to the dismal state of political science today. To begin with, there is no organizing paradigm or “model” of any kind. The field is often described as “mush.” At its best, the discipline serves up rules of thumb about alternative voting procedures and their relative desirability. Issues of incentives and incentive structure incompatibility are suppressed, even though they are as important in politics as in economics. Worse yet, the fundamental paradigm of politics is largely side‐stepped, namely “Politics: Who gets What, When, How,” as set forth in 1935 by Harold Laswell. That is, the all-important paradigm of politics as multilateral bargaining between interest groups is absent from the pages of most political science textbooks.
For reasons we are about to see, these deficiencies of contemporary political science must be remedied. In particular, we need a hard-core analytical model as compelling to Res Politica as the Law of Supply and Demand is to Res Economica.
B. Why the Paradigm of Economics No Longer Suffices
It is time to take a leaf from Aristotle, who correctly recognized that political science is the master discipline—not economics. Here are several reasons why:
First, by reviewing the meaning of “true capitalism” it is clear that our cherished paradigm of free market economics is completely dependent upon the assumptions of the rule of law, of unbribable judges, of sanctity of contract, and of transparency. Put bluntly: Proper political institutions are a necessary condition for the virtues of a free market system to deliver the outcome society wants. They come first. They are not an after-thought.
Second, the ability of a free market capitalist system to deliver the goods requires much more than the basic institutional set-up just described. Specifically, whenever issues of “public goods,” “externalities,” or “imperfect competition” arise, impacted interest groups must determine via multilateral bargaining exactly what gets provided, and who is to pay how much of the bill in the process. Moreover, in a global context, issues of how to cope with misaligned currencies, vast trade deficits, and theft of intellectual property rights will only be resolved politically via multilateral bargaining between myriad interest groups. This is part and parcel of a well-functioning capitalist system.
Third, we are living in a world where the price, quantity, and allocation of important commodities like oil were once determined by a free market. But they no longer are. We are now witnessing the ongoing and dangerous “politicization” of the oil, gas, copper, and other markets. The same is true in the case of multilateral bargaining over “intellectual property rights.”
Fourth and more broadly, most of the important issues that could stymie future world growth and precipitate war remain quintessentially political in nature. For starters: Who gets how much water at what price? Who will pay how much for global warming? How much will tomorrow’s youth be taxed to pay for the elderly? Which nations will be “allowed” to go nuclear? And how will rival claims in the Middle East eventually get sorted out?
In short, our future depends upon success in politics—that is, in the quality of future “governance” to utilize a preferable term. But what do we mean by “success in governance?” Is there a yardstick equivalent in politics to “resource allocation efficiency” in economics? More broadly, is there an organizing paradigm or model that could prove as useful to res politica in the future as the Law of Supply and Demand has proven useful to res economica in the past? Happily, there is. Yet this model is completely unknown to most political scientists and philosophers. This must change.
C. The Possibility of the Hegemony of Political Science
– The Nash-Harsanyi-Selten Pluralistic Bargaining Model –
The model in question is known as the Nash-Harsanyi-Selten (NHS) model of multilateral bargaining. It is one of the accomplishments that earned all three game theorists the only triple Nobel Prize awarded in economics (1994). Moreover, this model is one of the analytical marvels in the history of analytical science, and indeed of all science. (The fundamental paper in this regard is “A Simplified Bargaining Model for the n-Person Cooperative Game,” by John C. Harsanyi, International Economic Review, 4, pp. 194-220, 1963. This paper synthesizes and unifies the different theories of Nash, Selten, and Shapley into a coherent whole.) Before its development during the period of 1950–1965, concepts like “democratic pluralism,” “bargaining equilibrium,” “balance of power,” and “power” itself were problematically elastic concepts that lacked precise meaning. Additionally, without this model, the notion of relative bargaining ability could not be defined. For absent a model predicting an optimal bargaining equilibrium between symmetrically rational players, the degree to which one player bargained better than another could not be determined. By extension, it was impossible to assess the relative competence of different governments in striking bargains on behalf of their citizens without the yardstick such a model made possible.
The Building Blocks of the Model: The building blocks of the logic are starkly simple: (i) a set of n individual players; (ii) the set of 2n–2 possible coalitions that could form and oppose one another (e.g., the environmentalist lobby versus the lumber industry); (iii) the set of all n(n‐1)/2 possible pairs of players that could come face to face with each other in any number of coalitions that might include them both; and (iv) the different resources of each individual player and each coalition—including resources each could use to threaten the others.
The Bargaining Logic Utilized to Arrive at a Rational Compromise: In Stage 1 of the two-stage bargaining game, the various coalitions form and determine their best threat strategies to be utilized against their complementary coalitions in the event that no compromise ends up being reached, and the players fall back on playing their threat strategies (e.g., labor goes out on strike and/or management eliminates their jobs). In Stage 2, the all-player coalition of all n members forms, and its members determine how to allocate the gains to each player (above his threat payoff) that mutual cooperation makes possible.
The basic point is that, since everyone (with suitable side‐payments) can end up better off by compromising rather than receiving their non-cooperative threat payoff, they have an incentive to reach a compromise. This is, of course, the hallmark of all social life as we know it. In the NHS model, the compromise that rational players arrive at will be that agreement that equalizes the “risk limits” of every player, as John Harsanyi first pointed out. The interested reader is referred to a footnote that explains this remarkable result in more depth. (During the process of bargaining, each player starts off demanding more than he knows he will end up getting. As the game goes on, each player thus makes compromises so as to reduce the risk that others players say, “Screw you—we shall play our threat strategy against you!” Where does this process stop? What is the “sticking point” beyond which rational players will not make further concessions? It occurs at the point when the utility losses from making a further concession exactly equal the utility value of the reduction in risk that results from making the concession. [Mathematically, this point happens to be the outcome with the property that it maximizes the arithmetic product of the utility gains of the players above their threat payoffs. The product—not the sum!])
Market-based economic exchange is a very simple form of a bargaining game in which a consumer’s only threat strategy is simply not to buy a given product at the price offered. In more general political contexts, threats must be determined on the basis of how much damage a given coalition S (or single player) can do to its opposing coalition R net of the cost to itself S from carrying out its threat—relative to the damage the opposing coalition R can do to it S net of the cost to itself R from carrying out its threat. The logic is subtle: What matters is relative threat power.
The Remarkable Power of this Framework: There are four ways in which the NHS model is extremely powerful:
1. It Offers a Simple Graphical Representation of Politics: As stressed above, political science will never be a “science,” much less a successor to economics as a dominant paradigm without an intuitively appealing graphical model, such as that of intersecting supply and demand curves in economics. Happily, there does exist an analogous diagrammatic representation of bargaining. It is shown in the Appendix to this essay below.
2. It Incorporates the Right Mix of “Cooperative” and “Non‐Cooperative” Game Theory: In searching for the right paradigm with which to make sense of strategic interaction, game theorists during past decades believed that they needed to choose between two very different kinds of games: non-cooperative versus cooperative games. In the former case, emphasis is placed on the requirement that every player individually adopts a strategy that is optimal against every other individual’s strategy. This requirement must hold symmetrically for every player. Moreover, there are no coalitions in non-cooperative games.
In this paradigm, cooperation between people of the kind that arises in multilateral bargaining is oddly absent. The best, and indeed most celebrated, example of such a game is the Prisoner’s Dilemma in which, since neither prisoner can get together with the other and make a binding agreement not to tattle on the other, no gains from cooperation are possible. In this pathological case, the solution of the game (the non-cooperative Nash equilibrium) is for each prisoner in isolation to tattle on the other. The result: Each serves a much longer term in prison than would have been the case could they have communicated and agreed not to tattle.
Regrettably, this non-cooperative paradigm has dominated game theory for the past two decades. Previously, the cooperative paradigm had been dominant. In this latter case, the perspective is one in which players enter into groups for the purposes of adopting coordinated strategies that end up leaving everyone better off. In other words, they utilize outright bargaining to arrive at an optimal division of the spoils resulting from cooperation. Cooperation is central. The problem with most of these models was that they gave no play to the phenomenon whereby players adopt credible threats against one another as a prelude to the “final settlement.” In short, if classical non-cooperative game theory suffered from ignoring the gains from cooperation, classical cooperative theory failed to incorporate the non-cooperative aspects of human relations (threat-making in particular) in a proper manner.
It is one of the great virtues of the NHS model that it fully integrates both aspects of politics into a coherent model: The non-cooperative posturing (“If I don’t get my way, I’ll see that you pay dearly”) is integrated with the cooperative process of arriving at a final distribution of the proceeds from cooperation. It was John Harsanyi, who in 1963 provided a complete unification along these lines in games with n>2 players, and demonstrated mathematically how the two principal dimensions of bargaining (the threat game versus the cooperative game) are logically interdependent: One cannot be solved without solving the other. (Specifically, the equations characterizing the bargaining equilibrium are a set of simultaneous nonlinear equations, as is true of the general model of supply and demand in economics (Arrow-Debreu general equilibrium theory), and in many models within physics and biology.)
3. It Provides of a Yardstick for Measuring Bargaining Ability and thus Political Competence: As we suggested above, in the absence of a compelling definition of a rational bargaining outcome, it is difficult to say whether a given party bargained “competently” or “incompetently.” By extension, with no yardstick in hand, there can be little accountability by government to its citizenry regarding the quality of bargains it strikes, whether implicitly or explicitly. Happily, the NHS model provides precisely the missing yardstick. We will demonstrate this qualitatively in Section D just below where we apply bargaining logic to the difficult problem of negotiating with China.
4. It Offers a Unifying Framework for the Moral Tripos of Politics, Economics, and Ethics: We have already cited several of the reasons for the primacy of politics over economics (e.g., the importance of the rule of law as a precursor of market economies, as well as the bargaining that arises in dealing with market externalities, with public goods, with imperfect competition, and with trade and currency values). But when the NHS perspective is introduced, the potential hegemony of politics far transcends these issues of economics.
The physicist Mendel Sachs has recently identified a single truly unified field theory in physics from which all manifestations of matter (quantum phenomena, gravity, and electromagnetism) can be derived—just as Einstein always predicted would be the case (Sachs, M. Quantum Mechanics and Gravity, Springer Verlag, 2004). Analogously, and remarkably, it turns out that the NHS bargaining model can provide a unified framework for several disciplines within social science. In particular, the model can be “extended” in many different directions to re-derive the most serious theories now existing of interest group politics and of unbiased political representation and of perfectly competitive market economies and of the moral philosophical theory of Distributive Justice (astonishingly, “To Each according to His Contribution” and “To Each According to his Needs” can both be derived from the NHS model). The author is now writing a book on this subject.
D. An Application of the NHS Bargaining Model
– Case Study of How to Redress China’s Role in Global Imbalances –
The Consensus: In recent years and especially during recent months, the economics establishment has come down hard against those who believe it is time to retaliate against China—a view increasingly proposed by Democratic legislators and candidates for the US presidency. Whether it be Paul Krugman, Martin Wolf, David Hale, the editors of the New York Times or the Wall Street Journal or the Financial Times, the consensus of the intelligentsia is: “Neither the US nor the West, more broadly, should fall for the populist trap of protectionism. China needs time to develop, and must be encouraged to undertake a gradual approach to currency revaluation and reform in general.”
Other commentators go further and suggest that both parties are gaining from today’s status quo: “The US obtains cheap financing of its current account deficit, as well as products whose low Wal-Mart prices have kept inflation in check, whereas China obtains the huge market that it needs for its export machine.” This seductive argument runs afoul of the “no Free Lunch” theorem in economics. In the present case, this translates into the reality that the US will end up $4 trillion in debt to China for all these goodies—a debt we will pass on to our children, in addition to trillions of domestic debt.
The Fallacious Reasoning Underlying the Consensus View: Given China’s clearly undervalued currency and skyrocketing global trade surplus, what is the origin of the view that we must not fall prey to protectionism? Its origin is very interesting, and is central to the change of paradigm that we are proposing in this essay. The principal justification of the consensus is that, should we retaliate against Chinese policies via the imposition of tariffs, a trade war would result. For example, as a lead New York Times editorial of August 13, 2007 stated: “We have consistently argued against such punitive legislation, which could harm America’s economy by unleashing a trade war.”
An Alternative and More Constructive Perspective: But is this, in fact, the case? Need US legislation unleash a trade war? The answer to both questions is “No,” once a proper bargaining perspective is adopted. More specifically, today’s consensus is based upon the widespread assumption that resolving trade frictions constitutes a non-cooperative game. The logic that, if we do anything to upset China, they should and will retaliate, is taken from the logic of non-cooperative games like the Prisoner’s Dilemma. But this is not the correct logic, especially since the very concept of economic exchange is cooperative in nature. When I sell to you and you buy from me, we must both be gaining or else the trade would not have occurred. Thus, we need to adopt a cooperative game perspective—but one in which the role of mutual threats and recriminations assume their proper toll. This is exactly where the NHS perspective rises to the fore.
Here is what this perspective says regarding bargaining with China at present:
First, recognize that the concept of retaliation as being “protectionist” is nonsensical when China (and Asia, more broadly) admits to having been mercantilist for decades. We reiterate a point made in previous reports: Under true capitalism, there could not exist a $2 trillion cumulative US trade deficit with China, much less a cumulative $4 trillion deficit with Asia as a whole. For under true capitalism (no mercantilism, open capital accounts, transparency, and market determined currencies), these figures would be in the realm of $0 as a long-term average. Thus, for the victims of mercantilism to eventually rise up and protect themselves is not “protectionist” in any meaningful sense. Rather, it is a rational response to their victimization over decades.
Second, understand that the wrong response would be a piecemeal implementation of specific industry-by-industry tariffs on a nation-by-nation basis. Regrettably, this is the uncoordinated strategy that is now being adopted.
Third, implement the right strategy as dictated by the logic of true multilateral bargaining theory. This would be a coalitional strategy implemented by all nations victimized by Asian mercantilism—a strategy taking proper advantage of all their coalitional muscle and threat power. More specifically, China (and certain other nations) should be told:
“We want you in the World Trade Organization (WTO). We welcome your economic ascendancy and the opportunity to trade with you. We are not going to offend you by imposing willy nilly tariffs on a case-by-case basis. However, you promised ten years ago that you would curtail trade in stolen goods and patents, yet your export of these has more than doubled since 2002. You also promised to open your capital account and deregulate your financial system, but progress has been extremely slow. Finally, you agreed that your trade deficit would be curtailed (primarily through a significant appreciation of the Yuan), but on a trade-weighted basis, the Yuan has not risen, and your trade surplus has mushroomed. This state of affairs cannot and will not go on.”
“While we are not going to retaliate tomorrow morning on a case-by-case basis, we are, as a group of nations that uphold the covenants of the WTO, now going to insist that as regards intellectual property rights and counterfeit goods, you have two years to achieve a —% reduction; as regards your closed financial system, you have three years to implement policies A, B, and C respectively; and as regards your trade surplus and your undervalued currency, you have five years in which to achieve a —% reduction in your surplus with us, and five years to bring about a —% increase in the trade-weighted value of the Yuan.”
“Should you continue to stonewall such reforms, then in two years, all of us will impose a 35% quota on all goods we buy from you. In four years, this will rise to 70%. And in five years it will rise to 100%. This is not an idle threat. We are joined in common purpose here to help you and to redress our own problems. If you do not cooperate and refuse to change, the impact of our joint strategy will be to reduce your growth rate from 9% to an estimated 5%—a growth rate that will put tens of millions of your workers out of work, create social instability, and threaten your entire banking system. Please join us in working out this problem so that we might all come out ahead. We have no desire to damage your economy. However, history makes all too clear that, the longer these excesses and imbalances go on, the worse the ultimate denouement for all. Today’s status quo must thus end.”
Providing Political Cover to the Chinese Government: Note how this strategy is deliberately sympathetic to the position of the government of China. After all, just as is true in the States, the government of China itself is beholden to special interest groups and is thus quite weak. That is why a coordinated strategy by all of China’s trade victims, giving China several years to comply, is essential. It offers the government of China political cover to persuade their internal interest groups that, “This time, the West means business, and their threat is both powerful and credible. We have to give them part of what they want.” Note how different the strategy we have outlined is from that which is now likely to be implemented—a piecemeal tariff by one industry in September, another in November, etc. Each will invite a minor retaliation by China— a “tit for tat” strategy of the kind analyzed in non-cooperative game theory, precisely the wrong model to use in this essentially cooperative context.
China’s Counter-Threat: Game theory, like relativity theory in physics, is based upon symmetry: The views of both sides must symmetrically enter the picture. How should China respond to the above strategy by its trade partners? To begin with, the government of China will not welcome this message, and will issue counter-threats, just as it did during the week of August 6, 2007. Xia Bin, head of financial research at a key government think tank, said that China should use its gargantuan holdings of foreign exchange reserves in US treasuries as a “bargaining chip” (his words) in bilateral negotiations with the US. Then, He Fan of the Chinese Academy of Social Sciences, writing in The China Daily on August 9, warned that China may (for political reasons) be forced to sell large holdings of dollars leading to a mass depreciation of the US dollar.
But is this threat credible? This is the question that John Nash, Jr. taught us to address back in 1953 by analyzing the relative threat power of both sides, and then identifying the unique “mutually optimal and credible threat strategies between the players” that he proved will always exist. In the present case study, for China to dump US securities would indeed precipitate a large drop in the dollar. Yet Chinese ministers have told this author privately, “We have been, and will hopefully continue to be, semi-pegged to the dollar… If we were to cause the dollar to tank, then our currency relative to those of many of our other trading partners (e.g., all of Europe) would fall, further infuriating them against us. This is hardly an outcome we would welcome.” But above and beyond this point, the threat of the West to drive down China’s growth rate would indeed imperil the nation’s banking system—and this is a very powerful and credible threat indeed against China. And they know this.
Now, consider the price paid by the coalition of nations that might end up imposing significant tariffs on China to force it to adhere to WTO conventions. Yes, their domestic inflation rate would rise a bit as Chinese imports would be more expensive and/or less available. Yet on the other hand, domestic production of certain goods would increase, thus stimulating the growth rate of the economies involved. To restate this more analytically, the resulting reduction in their trade deficit with China would be offset by a corresponding increase in their GDP growth, via the elementary relationships of National Income Accounting.
Judging the Quality and Competence of Bargaining: Given these numerous observations, made possible by applying the NHS perspective to this textbook case of dealing with China, we are forced to ask how the Establishment has adopted the supine position that it has, namely “Don’t rock the boat…Give China time…Don’t start a tit-for-tat trade war by taking action…And don’t irritate China—it can sell its dollar assets and cause the dollar to collapse!” The answer is that most observers think incorrectly about the nature of bargaining. They think at best in terms of the non-cooperative model, and at worse they do not think at all. The NHS model is the correct model and prescribes a set of strategies diametrically opposite to those we have implicitly adopted.
Regrettably, it is the Chinese who do understand the logic of bargaining, and they exploit it brilliantly. So does Putin, who has exploited it brilliantly in his craven dealings with nations dependent upon Russian gas and oil. All of this is to say that the West is the classic frog being unwittingly parboiled to death by incompetent governance. Bluntly, our politicians should receive a very low score on their “bargaining competence test.”
Yet absent the fundamental shift in paradigm proposed in this essay, there can be no yardstick with which to hold governments responsible for such a squandering of our future.
APPENDIX – A Diagrammatic Representation of the NHS Model
In this Appendix, we set forth the basic logic of the bargaining problem as conceived by Nash, Harsanyi, and Selten.
The purpose of this brief Appendix is to show that, just as any beginning student of economic learns about supply and demand curves, and how their “intersection” yields market‐clearing prices and quantities of commodities, any beginning student in political science can and should learn the elementary graphical model that is set forth below. Indeed, this should provide the basis for a serious new course in what used to be called “civics.”
– Simple Two-Person “Pie Division” Case with No Optimal Threat Selection –
This is the simplest of all cases. Two players must determine how to divide a pie by noon tomorrow. If they do not reach an agreement, then both leave the table empty handed. Determining the disagreement payoff in this simple case does not entail any selection of optimal threats as it does in the general case. Rather, it is simply the “no agreement = no pie” payoff.
The Two Utility Curves: We start off with the concept of each player’s preferences for all possible allocations of pie, that is, their “utility functions” for pie. These preferences are shown in Figure 1. By convention, a utility score of 0 is assigned to each player for the worst outcome (no pie), and a score of 1 is assigned to the best outcome (all the pie). The numbers in between 0 and 1 are determined by an assessment procedure well-known in economics. Basically, the more curved (“concave”) a player’s utility function is, the more utility he attaches to the first piece of pie compared to the second, and the more he attaches to the second than the third, etc. This is the property of “declining marginal utility for pie.” In Figure 1, we show a traditional curve of this kind for player 1, and contrast it with the straight line constant marginal utility curve for player 2. (Interestingly, the degree of concavity of these curves can be interpreted in a manner completely different from that of the degree of decreasing marginal utility of a player. The alternative interpretation is that of the degree of the player’s relative risk aversion. The more concave the function is, the more risk averse the player is. A straight line utility function represents the limiting case of zero risk aversion, or “risk neutrality.”)
The “Prospect Space” of the Bargaining Game: Figure 2 then plots the two utility payoffs corresponding to all possible divisions of the pie. This set of all possible payoffs constitutes the curved “utility frontier” or boundary in the figure. The diagram is self explanatory: For any possible pie division, such as ••• of the pie for player 1 and ••• for player 2, just read off the corresponding utility payoffs to each from their respective utility curves shown in Figure 1. This doublet of utility payoffs then gets plotted as one point on the boundary, as in the two pictorial examples shown. The entire frontier is constructed in this manner, with any possible division of the pie corresponding to exactly one point on the frontier.
The Solution—Who Gets How Much Pie: The NHS solution is astonishingly simple, even if the logic underlying it is very subtle. If both players are equally rational, then they will end up dividing the pie such that the arithmetic product of the two associated utility payoffs is maximized. Note: Not the sum, but the product! In the example of Figure 2, the solution is the pie division (not shown) generating the payoff doublet U* = (.6, .7). To see visually that this particular point on the frontier maximizes the product of the utilities, note that its coordinates define the box with the largest possible area within the prospect space. More intuitively, this turns out to be the particular division of the pie at which each player’s “risk limits” are equalized in the process of mutual concession-making central to bargaining. It is the point where it becomes irrational for any player to make a further concession to the other, for the reasons described in Footnote 1 above. Intuitively, the player with the more concave (curved) utility function gets less, because he is “more needy” (or equivalently “more risk averse”) and thus gets bargained down by his opponent. (The mathematical equivalence of “relative risk averseness” and “relative neediness” was proven by the author. See “To Each According to His Needs: An Axiomatic Characterization,” appearing in Studies in Economic Theory, Volume 18, Edited by C. D. Aliprantis, Kenneth J Arrow, and Peter Hammond, Springer Verlag, 2004.)
Extension to n>2 Players, with Optimal Threat Selection: There are generally two stages in a bargaining game, whether with two players or more. In Stage 1, the players determine their optimal threat strategies against one another. Then in Stage 2, they attempt to reach a compromise from which everyone does better than by playing their threats. Figure 3 sketches this fully general solution, and demonstrates how it is a straightforward extension of the simple case shown above.
Once again, we always let 0 denote the utility payoff to every player from the worst possible outcome, and 1 the payoff for the best. Note here that players 1 and 2 have been replaced by players i and j to make the point that this is an n>2 person game. The point t* is known as the “net threat payoff vector.” Its coordinates, projected onto each of the n utility axes, represent the utility payoff to each player if and when all coalitions play their threats against one another—threats that must be determined according to the Stage 1 logic of relative threat power summarized in the text.
Then in Stage 2 of the game, all n players determine how to reach a compromise relative to this lurking threat payoff. In doing so, they arrive at the solution that maximizes the product of the utility gains for each player above and beyond his threat payoff, i.e.,
(1) MAX ∏ (Ui – t*i ) i= 1
Thus the logic of the simple two-person pie division carries right over to the complex case, the principal modification being the need to determine the “reference point” (the threat payoffs) that “orients” the Stage 2 bargaining game. All this should be clear from Figure 3. It should also be clear that, the further to the right the threat payoff is, the more pie the game will award to player j. To conclude, a given player will do better than another player if (i) he is less risk averse, and (ii) he has greater threat power as a result of the coalitions he is in. This is exactly how threat power ends up mattering to the final solution. In the case shown, it gives player j an advantage.
The purpose of this example has been to point out that via two quite simple diagrams, the essence of the NHS model of politics can be explained in an intuitively appealing visual manner, just as the law of supply and demand is in economics. The larger purpose of the foregoing essay was to convince the reader of the need for, and the timeliness of this bargaining theoretic paradigm. For in the new century we have entered, issues of politics and good governance must trump those of economics alone.
Source: JohnMauldin.com (