Game Theory An assessment of the theory that promises to unify the social sciences

1. Game TheoryAn assessment of the theory that promises to unify the social sciences YanisVaroufakis Visiting Professor at lbj school of public affairs Professor of economic theory at the university of athens

2. GT’s five central theorems • All (finite) interactions among autonomous agents have a rational ‘solution’ • All negotiations feature a uniquely rational bargaining agreement • Theorems 1 and 2 extend to dynamic settings • Theorems 1 and 2 extend to risky choices • No rationality required!

3. THEOREM 1: All finite non-cooperative interactions have a rational ‘solution’ (=equilibrium) Nash equilibrium A set of mutually reinforcing best replies

4. THEOREM 1: All finite non-cooperative interactions have a rational ‘solution’ (=equilibrium) • Nash equilibrium: • A set of mutually best replies • A set of strategies that confirm the beliefs that motivated them

5. THEOREM 1: All finite non-cooperative interactions have a rational ‘solution’ (=equilibrium) EXISTENCE THEOREM There exists at least one such Nash equilibrium in all possible interactions as long as: • the set of strategies is bounded • agents are labouring under commonly known instrumental rationality

6. Three possible critiques • Normative: Rationality means more than an incapacity to resist a net individual utility gain • Predictive: Real people are not that good at calculating (bounded rationality) • Immanent Criticism: Are the theory’s claims consistent with its assumptions?

7. Normative critique of instrumental rationality 1: A public good interaction • Group of N anonymous persons • Each is given $10 • Then, each is given two options • Option 1: Keep the $10 • Option 2: Contribute it to a common purse • Once all N persons have chosen, the contents of the purse are doubled (by the experimenter) and shared • Social optimum: All contribute and get $20 each • Nash equilibrium: No rational agent will contribute! • Martin Hollis: A scandalous conclusion

8. Normative critique of instrumental rationality 2: A riddle • Mary attends her mother’s funeral. • There, she spots a handsome man and develops an urge to talk to him, with a view to asking him out. • However, as the funeral comes to a close, the handsome man disappears without Mary managing to strike a conversation. • That night, Mary murders her sister. • Why did she do it? • Of 173 game theory students asked, 124 answered: “To see the handsome man again.” • Of 564 non-game theory students asked, only 32 offered the same answer

9. Predictive critique: Bounded rationality • Collective irrationality: Are you a better or worse driver than average? • Individual irrationality: HIV test • Suppose that 100000 random tests have been conducted • Suppose that, on that basis, the test for HIV proved accurate 99% of the time • Suppose also that 0.1% of the population have HIV, on average • You test positive. What is the probability that you have HIV? • Answer: <10%

10. THEOREM 2: A unique solution for the bargaining problem Bargaining protocol Round 1 A proposes division (x1,1-x1) B accepts END B rejects & Counter-offers (y1,1-y1); x1 >y1 Also, he selects probability of no agreement = 1-p1 Round 2 A accepts END A rejects Counter-offers (y2,1-y2); x2 <y2 and selects probability of no agreement = 1-p2 Rounds 3,4, ad infinitum

11. Defining credible rejections Suppose at round t A demands xt and offers B 1-xt A rejection by B, followed by a counterproposal of division (yt+1,1-yt+1) and a threat of conflict equal to pt is defined as credibleif and only if B expects a net gain from this rejection. In short: ptuB(1-yt+1)>uB(1-xt) Similarly, A’s rejection of B’s offer at t+1, yt+1, in favour of some other demand, say xt+1 , is credible if and only if pt+1uA(xt+1)>uA(yt)

12. Nash’s Remarkable TheoremThere exists only a single such agreement in every conceivable bargaining situation An equilibrium of fear (of disagreement) • Consider division/agreement Z • Suppose that if A offers Z to B, B can credibly reject Z • But, at the same time, A can credibly reject any such rejection. • Suppose that the same applies if B offers Z to A. • Then, Z is defined as an equilibrium of fear agreement – an EFA The one that maximises the product of their utilities!T

13. Theorem 3: First two theorems extend to dynamic settings • 7 $1000 bills on a table • A and B take turns to collect either 1 or 2 of them at each visit • The moment a player picks up 2, the game ends • Social optimum: Each takes one at each visit until no money is left on the table • Nash’s equilibrium: First player collects 2 bills, the second does not get to play and 5 bills are lost

14. Theorem 4: Theorems 1,2&3 extend to risky choices • Suppose A is ‘righteous’ (or Kantian) with probability p • Does it make sense for an instrumental A to pretend to be righteous/Kantian? • Yes, says Game Theory under the condition that… • the probability of righteousness is commonly known amongst all non-righteous players • the probability of bluffing (by non-righteous players who are pretending to be righteous) is also common knowledge amongst the non-righteous

15. Theorem 5: Instrumental Rationality is not even necessary! • Agency shifts from humans to… strategies (e.g. the strategy of Richard Dawkins’ ‘selfish gene’) • Pure adaptation of strategies • Plus random mutations • The strategy that does ‘better’ displaces the rest • Two mechanisms • An adaptation mechanism (replicator dynamic) • A mutation generating mechanism (testing the evolved strategy’s stability)

16. Theorem 5: Evolutionary Game Theory • Fundamental Theorem of Evolutionary Game Theory • Every evolutionary equilibrium is a Nash equilibrium • The opposite does not hold!

17. Immanent Criticism/internal critique 1. Too many equilibria: Indeterminacy The more realistic the settings, the more radical indeterminacy becomes

18. Immanent Criticism/internal critique 2. Unconvincing equilibria(Non-convergence even in logical time). The static case

19. Immanent Criticism/internal critique A ‘trivial’ bargaining solution? • When a player rejects an offer x, and threatens conflict with probability 1-p, Nash implicitly assumes common knowledge of 1-p, the probability of conflict • Over time, this is equivalent to assuming common knowledge of the expected duration of the negotiations • The uniqueness of the solution is thus assumed – not proven. • Nash’s proof: ifthere is a unique solution, it is the one that maximises the product of utilities.

20. Immanent Criticism/internal critique Trivial dynamic analyses? At every juncture common knowledge of subjective probability beliefs in the context of backward induction… • An epistemological minefield… • A monologue of (instrumental) Reason on Unreason and on alternative forms of rationality so as to keep players on the ‘equilibrium path’ • Denial of human reason’s capacity to subvert itself

21. Immanent Criticism/internal critique Recapping • An impoverished notion of Reason (instrumental rationality) unable to shoulder the explanatory burden • Closure demands the radical absence of a theory of motivated errors: Players are denied even their instrumental rationality for the glory of Game Theory • In its evolutionary guise, Game Theory presumes uncorrelated mutations The End of Politics and History…

22. Game Theory’s Conundrum • Models that are either over-determined or under-determined and can only be closed by sleight of hand • Is this surprising? • Why have so many brilliant minds proven so uncritical in their acceptance of this research program?

23. The Game Theorist’s Nemesis • Constantly elevating problems onto higher planes of abstraction, without solving them • Constantly introducing less and less credible axioms for the purpose of imposing an equilibrium, with little effect: In the end, Indeterminacy always rears its ugly head…

24. My end-of-the-road ‘discoveries’ • Game theory brings to its logical conclusion the theoretical project of explaining the world in terms of instrumental rationality • It reaps the harvest it has sown • Instrumental Reason has, and can have, no monopoly of insights on non instrumental behaviour • Errors cannot be segregated from rational attempts to subvert logic • When assuming that such segregation is possible (as GT does), theory ends up in a logical mess

25. The thoughts which Game Theory cannot fathom… • It overestimates our computing power and underestimates our Reason • Albert Camus: “Man as the only creature who refuses to be what he is.” • Hegel: “It is only by being acknowledged that I can be sure that I am, and of what I am.”

26. The Game Theorists’ strategy • Argue that Game Theory is the common language that can help unify the social sciences • Construct sophisticated interactions • Behind the scenes impose equilibria in order to ‘close’ the models • Destroy all sophistication in the process • Maintain academic power by putting on display either indeterminate rich analyses or determinate poor analyses – concealing that it is impossible to have both

27. The young Game Theorist’s Tragedy • Attempts to civilise/socialise homo economicus lead to radical indeterminacy • Psychological Game Theory • When we care not only about what others will do but also about the reasons for which they will do it • 2nd order beliefs infiltrate the player’s utility function

28. The young game theorist’s dilemma A truly illiberal move • To publish, one needs to ‘close’ the model • To close the model, the sophistication must go and intellectually indefensible hidden axioms must be introduced. The Dilemma • ‘Close’ it illicitly and lose its sophistication, as well as any prospect of telling a useful story about truly rational people • Refuse to do so and be damned…

29. Four conclusions • GT refuses to engage in the contradictions that are endogenously generated in its models • GT is useful in exploring the limits of anysocial theory - especially of audacious ones reaching for the Theory-of-Everything-Social status • GT’s most peculiar grand failure is the ultimate guide to the limitations of any attempt to understand human society in terms of liberal individualism • Behind every toxic derivative produced by Wall Street and every macroeconomic model backing the policies that led to 2008, lurks the Hubris of the game theorist’s strategy

30. Lastly… To engage in such discursive battles, graduate students must be prepared for a lonely struggle against a powerful Priesthood: “Azandesee as well as we that the failure of their oracle to prophesy truly calls for explanation, but so entangled are they in mystical notions that they must make use of them to account for failure. The contradiction between experience and one mystical notion is explained by reference to other mystical notions.” (Evans- Pritchard in his Witchcraft, Oracles and Magic among the Azande, 1937)

31. The debris of such a path… • Rational Conflict, Oxford: Blackwell Publishers, 1991 • Game Theory: a critical introduction, Routledge, (with Shaun Hargreaves-Heap), 1995 • Game Theory: Critical Concepts Vol. 1-5, Routledge • Game Theory: a critical text, Routledge, (with Shaun Hargreaves-Heap), 2004 • Economic Indeterminacy: An encounter with the economists’ peculiar nemesis, Routledge, 2013

32. The elusive arithmetic of Bayes’ Rule (99%) true +ve = 99 (100) (1%) HIVfalse –ve = 1 (0.1%) 100000 Tests false +ve = 999 no HIV (99900) true -ve = 98901 Pr(HIV│you tested +ve) = 99/(99+999) < 10%!