quantum probabilistic description of dealing with risk and ambiguity in foraging decisions

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

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


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.



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.



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.



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.



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.



Outline



3 Simulation

4 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.



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.



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.



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.



Outline



3 Simulation

4 Conclusions



The Simulated Environment

Forager

Patch of resource



Decision by the agent

Currentstate

Move West Move North Forage

For each A0

Evaluate possible A1

A1

A0


A2


For each A1

Evaluate possible A2



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.


https://github.com/xrubio/pandora


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.



Results – gradual resource map

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0.1

0.2

0.3

0.4

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 = 1

(a) Horizon=1

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0.2

0.4

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

ood

inta

ke

horizon = 3

(b) Horizon=3

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0.2

0.4

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

ood

inta

ke

horizon = 5

(c) Horizon=5

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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

ood

inta

ke

horizon = 7

(d) Horizon=7

Figure : Net food intakes for different scenarios using the gradualresource map



Results – random resource map

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0.1

0.2

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0.4

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 = 1

(a) Horizon=1

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0.2

0.4

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

ood

inta

ke

horizon = 3

(b) Horizon=3

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0.2

0.4

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

ood

inta

ke

horizon = 5

(c) Horizon=5

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0.2

0.4

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

ood

inta

ke

horizon = 7

(d) Horizon=7

Figure : Net food intakes for different scenarios using the randomresource map



Outline



3 Simulation

4 Conclusions



Limitations

Markovian decision function is not accurate;Determining the constant in the uncertainty relation;Resource maps are not realistic.



Outlook

Searching in semantic memory;Short-term exploitative competition of stock traders;Social foraging;Consumer decisions;Neurological background to sequential/simultaneousdecision making.



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.


quantum probabilistic description of dealing with risk and ambiguity in foraging decisions

Technology

patches decisions

foraging parameter

conclusions peter wittek

sequence of optimal

patch risk

quantum probability

risk ambiguity

xavier rubiocampillo3