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Quantum Economics and Finance: the Quantum Option David Orrell

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Quantum Economics and Finance: the Quantum Option David Orrell Quantum Economics and Finance: The Quantum Option David Orrell Introduction Quantum economics and finance uses quantum mathematics to model phenomena including cognition, financial transactions, and the dynamics of money and credit This talk develops a framework for modelling financial decisions that is based on quantum probability We sketch a model for pricing of financial options that includes both cognitive effects, and the dynamics of supply and demand Outline: - Quantum probability - Applications in finance and cognition - Quantizing transactions - Option prices with supply and demand Quantum probability Niels Bohr: “If quantum Richard Feynman: “I think I John von Neumann: “You don’t understand quantum mechanics hasn’t profoundly can safely say that nobody mechanics, you just get used shocked you, you haven’t understands quantum to it” understood it yet” mechanics” https://uwaterloo.ca/institute-for-quantum-computing/quantum-computing-101 Modelling propensity Superposition 0 = ȁ1ۧ 1 Superposed state 1 1 1 1 = Probability of Tails is the ȁ0ۧ + ȁ1ۧ squared amplitude of T1 2 2 2 1 1 = ȁ0ۧ 0 Probability of Heads is the squared amplitude of H1 Classical: 1-norm, options are heads OR tails Quantum: 2-norm, superposition state, heads AND tails Negative probability −1 1 1 −1 = ȁ0ۧ + ȁ1ۧ 2 2 2 1 Need to allow complex coefficients for roots Switching from a 1-norm to a 2-norm leads to concepts of superposition, negative probability, interference, and entanglement Interference 1 1 1 퐻 = 2 1 −1 1 1 1 퐻 = 2 1 0 Hadamard transformation rotates clockwise by 45 degrees The superposed Heads/Tails coin becomes Heads Entanglement Two coins requires a 4-dimensional graph Coefficients can also be complex Quantum coins can be entangled, so e.g. only have HH or TT Quantum computers are based on qubits which are in a superposed state ۄor ȁ1 ۄmeasured as ȁ0 ,ۄQubits usually initialised to ȁ0 IBM Q 53 qb Logic gates are represented by unitary matrices which act on qubits 0 1 푋 = NOT X 1 0 0 1 1 0 ۄȁ1 = = = ۄ푋ȁ0 1 0 0 1 1 1 1 퐻 = Hadamard H 2 1 −1 1 1 1 1 1 1 1 ۄȁ1 + ۄȁ0 = = = ۄ퐻ȁ0 2 1 −1 0 2 1 2 cos 휃 −sin 휃 푅 = Rotation 푅 휃 sin 휃 cos 휃 휃 cos 휃 −sin 휃 1 cos 휃 ۄsin 휃 ȁ1 + ۄcos 휃 ȁ0 = = = ۄ푅 ȁ0 휃 sin 휃 cos 휃 0 sin 휃 Multiple qubits 1 푏1 푎1푏1 푎1 1 1 푎1 푏 푏 푎 푏 2 1 = 2 = 1 ⊗ 0 = ⊗ = ۄȁ00 0 0 0 푎2 푏2 푏1 푎2푏1 푎2 0 0 푏2 푎2푏2 1 0 1 = ⊗ = ۄȁ01 0 1 0 0 0 1 1 0 = ⊗ = ۄȁ10 0 0 1 Computational 0 basis vectors 0 0 0 0 = ⊗ = ۄȁ11 1 1 0 1 The C-NOT gate flips the state of the lower qubit depending on the state of the upper control qubit Can be used to create an entangled state 1 0 0 0 0 ۄȁ00 = ۄ푋 = 1 0 0 푋 ȁ00 푐 0 0 0 1 푐 0 0 1 0 ۄȁ01 = ۄ푋푐ȁ01 ۄȁ11 = ۄ푋푐ȁ10 ۄȁ10 = ۄ푋푐ȁ11 A state 휓 is entangled if it does not factor as a tensor product: is not entangled, but ۄ푏ȁ1 + ۄ푎ȁ0 ⨂ۄȁ0 = ۄ푏ȁ01 + ۄ휓1 = 푎ȁ00 is entangled ۄ푏ȁ11 + ۄ휓2 = 푎ȁ00 The Toffoli gate flips the state of the upper qubit depending on the state of the two control qubits The C-NOT gate flips the NOT gate flips the qubit state of the upper qubit depending on the state of the lower control qubit Some applications A useful circuit cos 휃 −sin 휃 푅 = 휃 sin 휃 cos 휃 Quantum money Early quantum computer Graphics from Instant Economics rotate 휃 = 45° might default RA collects RB rotate 휑 = 90° Creditor assumes no default Debtor defaults Creditor collects Probability Debtor defaults if top qubit is 1 Yes sin2 휃 = 0.5 No cos2 휃 = 0.5 Creditor paid if bottom qubit is 1 Yes sin2 휑 = 1 2 Both can’t happen at the same time! No cos 휑 = 0 C-NOT gate flips lower qubit depending on state of upper control qubit RA RB If the top qubit is 1 (so default), the bottom qubit is flipped to 0 Debtor defaults Creditor collects Probability Yes Yes 0 If the top qubit is 0 (so no default), No Yes cos2 휃 = 0.5 the bottom qubit remains as 1 Yes No sin2 휃 = 0.5 Financial entanglement! No No 0 To default, or not to default The debt circuit is quantum because the value of the debt depends on debtor who is modelled as being in a superposed, undecided state Can model cognition using the same circuit – decisions depend on subjective context John Barrymore as Hamlet (1922) Quantum cognition: the disjunction effect Experiment from Tversky and Shafir (1992): Students are told to imagine that they have a tough exam coming up, and have an opportunity to buy a vacation to Hawaii at a very good price – would they take the offer? In one version, they were told the result of the exam If the result was pass, 54% chose to buy If the result was fail, 57% chose to buy In another version, they were told they will not know the result In this case only 32% chose to buy Context creates mental interference! Uncertain Equal chance Adjust for of pass or fail chance to buy RA Single qubit represents state Vacation Probability Yes cos2 휃 − 휑 = 0.33 Test result is not measured No sin2 휃 − 휑 = 0.67 What if we want to measure the test result first? Independent RA Test yes/no Buy yes/no RB Top qubit represents probability of Test Vacation Probability passing or failing the test Pass cos2 휃 = 0.50 Fail sin2 휃 = 0.50 Bottom qubit adjusts for probability Yes cos2 휑 = 0.97 of decision to buy vacation No sin2 휑 = 0.03 Coupled RA RB Entangles context with decision Test Vacation Probability Pass Yes cos2 휃 cos2 휑 = 0.48 Used rotations, but works for any Pass No cos2 휃 sin2 휑 = 0.02 pair of quantum gates Fail Yes sin2 휃 sin2 휑 = 0.02 Fail No sin2 휃 cos2 휑 = 0.48 Quantum game theory: the prisoner’s dilemma A strategy B strategy A sentence B sentence silent silent 1 1 silent testify 3 0 testify silent 0 3 Nash equilibrium testify testify 2 2 Cold War The mathematician John von Neumann, who invented the theory of expected utility (as well as quantum logic), served as advisor to President Eisenhower on the use of the bomb He used this result from game theory to argue in favour of the first strike doctrine After the Soviets developed their own weapons, the strategy morphed to one of mutually assured destruction – or MAD Binary logic led us to the brink of nuclear war – mathematical models scale up! Experiments: 10% defect if other person’s strategy is known, rises to 37% if not known Without entanglement Other strategy RA Own strategy RB Top qubit A represents a person’s subjective beliefs about what another player will do Bottom qubit B represents the person’s own strategy With entanglement RA RB Experiments: 10% defect if other person’s strategy is known, rises to 37% if not known Agrees with quarter law, which is an empirically-tested result from quantum cognition (Yukalov & Sornette, 2018) Quantum agent-based model MomentumRA FundamentalRB Entanglement for single agent This is your brain on neoclassical economics ContextRA Corpus callosum severed RationalRB Context can represent the state of mind of a debtor (credit circuit), a preceding event (disjunction effect/order effect), presumed beliefs about another person (prisoner’s dilemma), or any subjective factor Here it has no effect on the decision or the outcome This is your brain on quantum economics ContextRA Aha moment! RationalRB Right hemisphere “sees each thing in its context” (McGilchrist, 2009: 49) In model, the change in perspective is a consequence of the shift from utility to probabilistic propensity, which produces entanglement This is two brains on quantum economics Context Rational Context Rational This is a computer on quantum AI Kondratyev (2020) Non-Differentiable Learning of Quantum Circuit Born Machine with Genetic Algorithm What does entanglement mean? Modelling choice: follows because we represent mental states or strategies as a superposition (qubit), rather than a binary choice (bit) In finance entanglement represents a contract, in quantum cognition it typically represents the connection between objective and subjective factors, in game theory it is used to model things like societal norms One paper concludes that quantum games “produce counter-intuitive outcomes such as a greater prevalence of altruism than in the classical game” ??? Wave splitter Decision Context Sequential A+, A− Entangled B+, B− Controlled gate plays role similar to a sequential IF-THEN statement, but without collapsing the qubit EVENTS 퐴+퐵+ 퐴+퐵− 퐴−퐵+ 퐴−퐵− Result State ȁ00ۧ ȁ01ۧ ȁ10ۧ ȁ11ۧ 2 2 2 2 Probability 푎11푏11 푎11푏21 푎21푏12 푎21푏22 But why quantum? Anti-cancer drugs show order effect (the order in which drugs are administered is very important), threshold effects (a minimum level of drug may be needed to have an effect), holism (connected system), interference (drug combinations can be synergistic or antagonistic), and are ultimately based on quantum processes (like everything else) but we don’t model them as quantum systems Reasons to be quantum Quantum finance: “The evolution into a superposition of financial states and their measurement by transaction is my understanding of quantum finance.” Schaden (2002) “It is the random time evolution of financial instruments that provides the crucial and far-reaching connection with quantum theory.” Baaquie (2004) “Every transaction in financial markets is in fact an elementary act of price measurement.” Sarkissian (2020) Quantum-like: “Our basic premise is that information processing by complex social systems can be described by the mathematical apparatus of quantum mechanics.” Haven & Khrennikov (2013) Quantum consciousness: “I argue that human beings and therefore social life exhibit quantum coherence – in effect, that we are walking wave functions.” Wendt (2015) Quantum computing: A quantum algorithm “could be deployed in days.
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