The Gödelian foundations of self-reference, the liar and incompleteness: arms race in complex strategic innovation. (English) Zbl 1407.91014
Pinto, Alberto A. (ed.) et al., Trends in mathematical economics. Dialogues between Southern Europe and Latin America. Selected papers based on the presentations at the conferences: 3rd international conference on dynamics, games and science, DGS III, on the occasion of the 50th birthday of Alberto A. Pinto, Porto, Portugal, February 17–21, 2014, the 1st Hellenic-Portuguese meeting on mathematical economics, AUEB, Athens, Greece, and the XV Jornadas Latinoamericanas de Teoría Económica, JOLATE, Guanajuato, México. Cham: Springer. 217-244 (2016); erratum ibid. E1 (2016).
Summary: Self-referential calculations of oppositional or contrarian structures and the necessity to innovate to outsmart hostile agents in an arms race are ubiquitous in socio-economic systems, immunology and evolutionary biology. However, such phenomena with strategic innovation, which entails novel actions beyond listable sets, are outside the ambit of extant game theory. How can strategic innovation with novel actions be a Nash equilibrium of a game? Based on the only known Gödel-Turing-Post (GTP) axiomatic framework on meta-analyses of offline simulations that involve recursive operations on encoded information, we show that mutually mentalising agents capable of such offline simulations can “think outside the box” and embark on an arms race in novelty or surprises. A key logical ingredient of this is the self-referential encoding of a proposition on mutual negation or opposition, often referred to as the Gödel sentence. The only recursive best response function of a two-person game with an oppositional structure that can implement strategic innovation in a lock-step formation of an arms race is the productive function of the Emil Post set theoretic proof of the Gödel incompleteness result.
For the entire collection see [Zbl 1364.91007].
For the entire collection see [Zbl 1364.91007].
Keywords:
Gödel incompleteness; self-reference; contrarian; offline simulation; strategic innovation; novelty; surprises; red queen-type arms race; creative and productive sets; productive function; surprise Nash equilibriumReferences:
| [1] | Albin, P.: The metalogic of economic predictions, calculations and propositions. Math. Soc. Sci. 3, 329-358 (1982) · Zbl 0503.90028 · doi:10.1016/0165-4896(82)90016-6 |
| [2] | Albin, P.: In: Duncan F. (ed.) Barriers and Bounds to Rationality, Essays on Economic Complexity and Dynamics in Interactive Systems. Princeton University Press, Princeton (1998) |
| [3] | Anderlini, L.: Some notes on Church’s thesis and the theory of games. Theory Decis. 29, 19-52 (1990) · Zbl 0697.68055 · doi:10.1007/BF00134103 |
| [4] | Anderlini, L., Sabourian, H.: Cooperation and effective computability. Econometrica 63, 1337-1369 (1995) · Zbl 0841.90129 · doi:10.2307/2171773 |
| [5] | Arthur, W.B.: On the evolution of complexity. Working Paper, 93-11-070. Santa Fe Institute, Santa Fe, NM (1993) |
| [6] | Arthur, W.B.: Inductive reasoning and bounded rationality. Am. Econ. Rev. (Pap. Proc.) 84, 406-411 (1994) |
| [7] | Arbib, M., Fagg, A.: Modeling parietal-premotor interactions in primate control of grasping. Neural Netw. 11, 1277-1303 (1998) · doi:10.1016/S0893-6080(98)00047-1 |
| [8] | Axelrod, R.: Risk in Networked Information Systems, Mimeo. Gerald R. Ford School of Public Policy, University of Michigan. (2003); Baumol, W.: The Free Market Innovation Machine. Princeton University Press, Princeton (2002) |
| [9] | Baumol, W.: The Free Market Innovation Machine. Princeton University Press, Princeton (2002) · doi:10.1515/9781400851638 |
| [10] | Baumol, W.: Red Queen games: arms races, rule of law and market economies. J. Evol. Econ. 14 (2), 237-247 (2004) · doi:10.1007/s00191-004-0207-y |
| [11] | Beinhocker, E.: Evolution as computation: integrating self-organization with generalized. J. Inst. Econ. 7 (3), 393-423 (2011) |
| [12] | Ben-Jacob, E.: Bacterial wisdom, Gödel’s theorem and creative genomic webs. Phys. A Stat. Mech. Appl. 248 (1), 57-76 (1998) · doi:10.1016/S0378-4371(97)00529-3 |
| [13] | Bhatt, M., Camerer, C.: Self-referential thinking and equilibrium as states of mind in games: FMRI evidence. Games Econ. Behav. 52, 424-459 (2005) · Zbl 1117.91302 · doi:10.1016/j.geb.2005.03.007 |
| [14] | Binmore, K.: Modelling rational players: Part 1. J. Econ. Philos. 3, 179-214 (1987) · doi:10.1017/S0266267100002893 |
| [15] | Byrne, R., Whiten, A.: (1999) Machiavellian Intelligence: Social Expertise and the Evolution of Intellect in Monkeys, Apes, and Humans. Oxford University Press, Oxford (1988) |
| [16] | Canning, D.: Rationality, Computability and Nash equilibrium. Econometrica 60, 877-895 (1992) · Zbl 0764.90091 |
| [17] | Casti, J.: Complexification: Explaining a Paradoxical World Through the Science of Surprises. London Harper Collins, London (1994) |
| [18] | Colander, D.: The Complexity Vision and the Teaching of Economics, Cheltenham (2000) |
| [19] | Cutland, N.J.: Computability: an introduction to recursive function theory, Cambridge University Press. expectations, learning and convergence to rational expectations equilibrium. Am. Econ. Rev. 72, 652-68 (1980) · Zbl 0448.03029 |
| [20] | Durlauf, S.: Complexity, economics and public policy. Pol. Philos. Econ. 11 (1), 45-75 (2012) · doi:10.1177/1470594X11434625 |
| [21] | Gallese, V., Sinigaglia, C.: What is so special about embodied simulation? Trends Cogn. Sci. 15 (11), 515 (2011) · doi:10.1016/j.tics.2011.09.003 |
| [22] | Gallese, V., Fadiga, L., Fogassi, L., Rizzolatti, G.: Action recognition in the premotor cortex. Brain 119 (2), 593-609 (1996). doi: 10.1093/brain/119.2.593. PMID 8800951 · doi:10.1093/brain/119.2.593. |
| [23] | Gödel, K.: On Formally Undecidable Propositions of Principia Mathematica and Related Systems [Translation in English in Gödel’s Theorem in Focus, ed. by S.G Shanker, 1988, Croom Helm] (1931) Goodhart C.: Central Bankers and Uncertainty, Keynes Lecture to the Royal Academy. Reprinted in the Bank of England Quarterly Bulletin, Feb, pp. 102-114 (1999) |
| [24] | Goodhart, C.: Central Bankers and Uncertainty, Keynes Lecture to the Royal Academy. Reprinted in the Bank of England Quarterly Bulletin, Feb, pp. 102-114 (1999) |
| [25] | Haldane, A.: Financial arms races, speech delivered at the institute for new economic thinking, Berlin, 14 April 2012 |
| [26] | Hayek, F.A.: The Sensory Order. The University of Chicago Press, Chicago (1952) |
| [27] | Hayek, F.A.: The Constitution of Liberty. University of Chicago Press, Chicago (1960) |
| [28] | Hayek, F.A.: The Theory of Complex Phenomena. Studies in Philosophy, Politics, and Economics. The University of Chicago Press, Chicago (1967) |
| [29] | Hirschman, A.O.: The Rhetoric of Reaction: Perversity, Futility, Jeopardy. The Belknap Press of Harvard University Press, Cambridge (1991) |
| [30] | Holland, J.: Adaptation in Natural and Artificial Systems. MIT Press, Cambridge (1975) · Zbl 0317.68006 |
| [31] | Kant, I.: Critique of Pure Reason (1965). Translated by Norman Kemp Smith. St. Martin’s, New York (1781) |
| [32] | Koons, R.: Paradoxes of Beliefs and Strategic Rationality. Cambridge University Press, Cambridge (1992) · doi:10.1017/CBO9780511625381 |
| [33] | Koppl, R., Rosser B.: Everything i might say will already have passed through your mind. Metroeconomica 53 (4), 339-360 (2002) · Zbl 1041.91017 · doi:10.1111/1467-999X.00147 |
| [34] | Langton, C.: Life at the edge of chaos. In: Langton, G.C., Taylor, C., Doyne Farmer, J., Rasmussen, S. (eds.) Artificial Life II (Sante Fe Institute Studies in the Sciences of Complexity, vol. 10). Addison-Wesley, Reading (1992) |
| [35] | Markose, S.M.: The new evolutionary computational paradigm of complex adaptive systems: challenges and prospects for economics and finance. In: Chen, S.-H. (ed.) Genetic Algorithms and Genetic Programming in Computational Finance, pp. 443-484. Kluwer Academic Publishers, Boston (2002). Also Essex University Economics Department DP no. 552, July 2001 |
| [36] | Markose, S.M.: Novelty in complex adaptive systems (CAS): a computational theory of actor innovation. Phys. A Stat. Mech. Appl. 344, 41-49 (2004). Fuller details in University of Essex, Economics Department. Discussion Paper No. 575, January 2004 |
| [37] | Markose, S.M.: Computability and evolutionary complexity: markets as complex adaptive systems (CAS). Econ. J. 115, F159-F192 (2005) · doi:10.1111/j.1468-0297.2005.01000.x |
| [38] | Markose, S.M.: Mirroring, Offline Simulation and Complex Strategic Interactions: Coordination, Anti-Coordination and Innovation. University of Essex Mimeo, Essex (2015) |
| [39] | Nachbar, J.H., Zame,W.R.: Non-computable strategies and discounted repeated games. Econ. Theory 8, 111-121 (1996) · Zbl 0852.90146 |
| [40] | Osborne, M., Rubinstein, A.: A Course in Game Theory. MIT Press, Cambridge (1994) · Zbl 1194.91003 |
| [41] | Post, E.: Recursively enumerable sets of positive integers and their decision problems. Bull. Am. Math. Soc. 50, 284-316 (1944) · Zbl 0063.06328 · doi:10.1090/S0002-9904-1944-08111-1 |
| [42] | Rizzolatti, G., Fadiga, L., Gallese, V., Fogassi, L.: Premotor cortex and the recognition of motor actions. Cogn. Brain Res. 3, 131-141 (1996) · doi:10.1016/0926-6410(95)00038-0 |
| [43] | Rogers, H.: Theory of Recursive Functions and Effective Computability. Mc Graw-Hill, New York (1967) · Zbl 0183.01401 |
| [44] | Sayama, H.: Construction theory, self-replication, and the halting problem. Complexity (2008). doi:10.1002/cplx · doi:10.1002/cplx |
| [45] | Schumpeter, J.A.: Capitalism, Socialism and Democracy, p. 139. Routledge, London (1994). ISBN 978-0-415-10762-4. (1942) |
| [46] | Simmons, K.: Universality of the Liar, Cambridge University Press, Cambridge (1993) · Zbl 0977.03502 · doi:10.1017/CBO9780511551499 |
| [47] | Smullyan, R.: Theory of Formal Systems. Princeton University Press, Princeton (1961) · Zbl 0097.24503 · doi:10.1515/9781400882007 |
| [48] | Spear, S.: Learning rational expectations under computability constraints. Econometrica 57, 889-910 (1989) · Zbl 0697.90009 · doi:10.2307/1913776 |
| [49] | Tesfatsion, L., Judd, K.L. (eds.): Handbook of Computational Economics: Volume2, Agent-Based Computational Economics. Handbooks in Economics Series, 904pp. Elsevier, North-Holland Imprint, Amsterdam (2006) · Zbl 1143.91300 |
| [50] | Tognoli, E., Lagarde, J., de Guzman, G.C., Kelso, J.A.S.: The phi complex as a neuromarker of human social coordination. Proc. Natl. Acad. Sci. USA 104 (19), 8190-8195 (2007) · doi:10.1073/pnas.0611453104 |
| [51] | Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. 2 42, 230-265 (1937) · JFM 62.1059.03 · doi:10.1112/plms/s2-42.1.230 |
| [52] | Wolfram, S.: Cellular automata as models of complexity. Physica 10D, 1-35 (1984) · Zbl 0562.68040 |
| [53] | Wolfram, S: A New Kind |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
