quantum probabilistic description of dealing with risk and ambiguity in foraging decisions
DESCRIPTION
A forager in a patchy environment faces two types of uncertainty: ambiguity regarding the quality of the current patch and risk associated with the background opportunities. We argue that the order in which the forager deals with these uncertainties has an impact on the decision whether to stay at the current patch. The order effect is formalised with a context-dependent quantum probabilistic framework. Using Heisenberg's uncertainty principle, we demonstrate the two types of uncertainty cannot be simultaneously minimised, hence putting a formal limit on rationality in decision making. We show the applicability of the contextual decision function with agent-based modelling. The simulations reveal order-dependence. Given that foraging is a universal pattern that goes beyond animal behaviour, the findings help understand similar phenomena in other fields.TRANSCRIPT
Quantum Probabilistic Description of Dealingwith Risk and Ambiguity in Foraging Decisions
PETER WITTEK1, Ik Soo Lim2, Xavier Rubio-Campillo3
1University of Boras
2Bangor University
3Barcelona Supercomputing Center
July 27, 2013
Foraging decisions Quantum probabilistic description Simulation Conclusions
Outline
1 Foraging decisions
2 Quantum probabilistic description
3 Simulation
4 Conclusions
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Foraging
A bumblebee worker finds a rewarding “flower.” Photograph by JayBiernaskie. From Biernaskie, J.; Walker, S. & Gegear, R. Bumblebeeslearn to forage like Bayesians. The American Naturalist, 2009, 174,
413–423.Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Optimal foraging theory
A successful approach in understanding animal decisionmaking;Assumption: organisms aim to maximise their net energyintake per unit time;Food sources are available in patches, which vary inquality;Switching between patches comes with a cost.
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Risk and ambiguity
Uncertainty: in decisions about staying at a patch ormoving on to the next one;Two fundamental types of uncertainty: ambiguity and risk;Ambiguity: the estimation of the quality of a patch;Risk: the potential quality of other patches;Decisions are quintessentially sequential.
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Bayesian strategy and ambiguity
Animals have an a priori assessments of food distributions;Foraging decisions are influenced by experience;Assumptions:
Perfect knowledge of patch-type distribution (a priori);No instant identification of the quality of a particular patch,resulting in a sample.
Foraging decision is formulated by an a posterioridistribution made using the sample;Density-dependent resource harvest;Second assumption corresponds to ambiguity.
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Optimal foraging theory and risk
Variations of any foraging parameter affect the expectedrate of food gain;Variance in a parameter is associated with risk;Empirical and theoretical proofs: foraging decisions arerisk-sensitive.Risk sensitivity should have a sequential component.
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Contextuality
“[C]ontext-dependent utility results from the fact thatperceived utility depends on background opportunities;”The sequence of optimal decisions depends on theattributes of the present opportunity and its backgroundoptions.Examples: Honey bees, rufous humming birds, gray jays,European starlings, etc.Humans alternate between sequential and simultaneousdecision making.
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Outline
1 Foraging decisions
2 Quantum probabilistic description
3 Simulation
4 Conclusions
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Probability space
Two hypotheses describe the decision space of a forager:h1: Stay at the current patch.h2: Leave the patch.
Consider the following events:A: Current patch quality with two possible outcomes: a1 –the patch quality is good; a2 – the patch quality is bad.B: Quality of other patches. A collective observation acrossall other patches with two possible outcomes.
A corresponds to ambiguity;B corresponds to risk.
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Probability space
A and B are incompatible observations on a system;Forager’s state of belief is described by a state vector;Under observation A, this superposition is written as
|ψ〉 =∑i,j
αij |Aij〉 (1)
The square norm of the corresponding projected vector willbe the quantum probability of h1 ∧ a1: ||P11|ψ〉|| = |α11|2.Observation B: the state of belief is a superposition of fourdifferent basis vectors: |ψ〉 =
∑i,j βij |Bij〉.
Under observation A, the forager bases its decision onlocal information;Under B, it looks at a global perspective.
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Context sensitivity
Luders’ rule;Suppose that event A is true, the patch is of good quality;Original state |ψ〉 changes to |ψA〉 = Pa1|ψ〉/||Pa1|ψ〉||;The probability of event B given that A is true:||Pb1|ψa1〉||2 = ||Pb1Pa1|ψ〉||2/||Pa1|ψ〉||2;Pa1Pb1 6= Pb1Pa1;Bayesian models have a difficulty accounting for ordereffects.
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Uncertainty in sequential decisions
A state cannot be a simultaneous eigenvector of the twoobservables in general;The forager needs to leave the current patch to assess thequality of other patches:
Inherent uncertainty in the decision irrespective of thequantity of information gained about either A or B.
With regards to risk and ambiguity, the uncertaintyprinciple holds:
σAσB ≥ c, (2)
Where c > 0 is a constant.
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Outline
1 Foraging decisions
2 Quantum probabilistic description
3 Simulation
4 Conclusions
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
The Simulated Environment
Forager
Patch of resource
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Decision by the agent
Currentstate
Move West Move North Forage
For each A0
Evaluate possible A1
A1
A0
Move West Move North Forage
A2
Move West Move North Forage
For each A1
Evaluate possible A2
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
The model
A small map with heterogeneous resources;An agent has requirements: the quantity of resourcesneeded at every time step;The agent has a limited time horizon;The agent has a knowledge of the patch distribution;The agent’s characteristics:
Ambiguity aversion: a value between 0 to 1 specifying thepreference of the agent to avoid ambiguity. Ambiguityaversion is related to observation A.Risk aversion: a value between 0 to 1 specifying thepreference of the agent to avoid risk. Risk aversion isrelated to observation B.
Possible scenarios are explored using a Markov decisionprocess;Source code:https://github.com/xrubio/pandora.
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
The decision function
Preference to minimizing risk (R) or minimizing ambiguity(A);Cost C of an action a given current state s is:
Cs,a =mM
+ (1− aA)A + (1− aR)R (3)
m is the resources consumed in this time step, Mcorresponds to the resource requirements, aA is thedecrease of ambiguity of the action, and aR is thedecrease of risk of the action.Two kinds of resource maps:
1 Gradual resource map. The value of each patch is its xrelative coordinate multiplied by 10.
2 Random resource map. The value of each patch is definedusing a random uniform distribution.
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Results – gradual resource map
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risk aversion
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horizon = 1
(a) Horizon=1
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horizon = 3
(b) Horizon=3
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horizon = 5
(c) Horizon=5
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horizon = 7
(d) Horizon=7
Figure : Net food intakes for different scenarios using the gradualresource map
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Results – random resource map
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
risk aversion
net f
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inta
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horizon = 1
(a) Horizon=1
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0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
risk aversion
net f
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horizon = 3
(b) Horizon=3
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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horizon = 5
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
risk aversion
net f
ood
inta
ke
horizon = 7
(d) Horizon=7
Figure : Net food intakes for different scenarios using the randomresource map
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Outline
1 Foraging decisions
2 Quantum probabilistic description
3 Simulation
4 Conclusions
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Limitations
Markovian decision function is not accurate;Determining the constant in the uncertainty relation;Resource maps are not realistic.
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Outlook
Searching in semantic memory;Short-term exploitative competition of stock traders;Social foraging;Consumer decisions;Neurological background to sequential/simultaneousdecision making.
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions
Foraging decisions Quantum probabilistic description Simulation Conclusions
Summary
Classical Markovian decision model producescontext-dependent characteristics in an agent-basedsimulation;Sequential decisions in foraging theory are well explainedin terms of Luder’s rule;Risk and ambiguity are incompatible concepts, leading toan uncertainty principle.Simultaneous decisions do not have such constraints.
Peter Wittek, Ik Soo Lim, Xavier Rubio-Campillo Quantum Probabilistic Description of Foraging Decisions