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Optimal Foraging Theory Revisited

Masters Thesis Defense and Doctoral Qualifier Examination^{∗}

Theodore (Ted) Pavlic

June 5, 2007

∗Typeset with L^{A}TEX using powerdot.

Overview

Introduction Solitary Agent Model Optimization Results Remarks Questions?

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Introduction

Solitary Agent Model

Optimization

Results

Remarks

Questions?

Introduction

Introduction Solitary Agent Model Optimization Results Remarks Questions?

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Goal

How to model?

How to measure performance?

High-level actuators are behaviors

Behavioral ecology replaces physics

Introduction Solitary Agent Model Optimization Results Remarks Questions?

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Goal

How to model?

How to measure performance?

High-level actuators are behaviors

Behavioral ecology replaces physics

Introduction Solitary Agent Model Optimization Results Remarks Questions?

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Goal: Discover the physics of high-level control

How to model?

How to measure performance?

High-level actuators are behaviors

Behavioral ecology replaces physics

Introduction Solitary Agent Model Optimization Results Remarks Questions?

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Goal: Discover the physics of high-level control

How to model?

How to measure performance?

High-level actuators are behaviors

Behavioral ecology replaces physics

Introduction Solitary Agent Model Optimization Results Remarks Questions?

3 / 24

Goal: Discover the physics of high-level control

How to model?

How to measure performance?

High-level actuators are behaviors

Behavioral ecology replaces physics

Introduction Solitary Agent Model Optimization Results Remarks Questions?

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Goal: Discover the physics of high-level control

How to model?

How to measure performance?

High-level actuators are behaviors

Behavioral ecology replaces physics

Introduction Solitary Agent Model Optimization Results Remarks Questions?

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Goal: Discover the physics of high-level control

How to model?

How to measure performance?

High-level actuators are behaviors

Behavioral ecology replaces physics

Solitary Agent Model

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

OFT Renewal Cycle

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Parameters and Decision Variables

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Parameters and Decision Variables

n: Number of task types.

λ_{i}: Poisson encounter rate with tasks of type i (encounters/second).

λ: Merged Poisson encounter rate with all tasks (encounters/second).

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Parameters and Decision Variables

n: Number of task types.

λ_{i}: Poisson encounter rate with tasks of type i (encounters/second).

λ: Merged Poisson encounter rate with all tasks (encounters/second).

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Parameters and Decision Variables

n: Number of task types.

λ_{i}: Poisson encounter rate with tasks of type i (encounters/second).

λ: Merged Poisson encounter rate with all tasks (encounters/second).

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Parameters and Decision Variables

λ = ∑_{i=1}^{n} λ_{i}

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Parameters and Decision Variables

c^{s}: Cost rate for searching (points/second).

\frac{c^{s}}{λ}: Average cycle search cost (points/encounter).

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Parameters and Decision Variables

c^{s}: Cost rate for searching (points/second).

\frac{c^{s}}{λ}: Average cycle search cost (points/encounter).

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Parameters and Decision Variables

p_{i}: Probability of processing a task of type i.

τ_{i}: Average time processing a task of type i (seconds).

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Parameters and Decision Variables

p_{i}: Probability of processing a task of type i.

τ_{i}: Average time processing a task of type i (seconds).

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Parameters and Decision Variables

g_{i}(τ_{i}): Average gain from processing a task of type i (points).

c_{i}: Average cost rate while processing a task of type i (points/second).

g_{i}(τ_{i}) − c_{i}τ_{i}: Average net gain from processing a task of type i (points).

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Parameters and Decision Variables

g_{i}(τ_{i}): Average gain from processing a task of type i (points).

c_{i}: Average cost rate while processing a task of type i (points/second).

g_{i}(τ_{i}) − c_{i}τ_{i}: Average net gain from processing a task of type i (points).

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Parameters and Decision Variables

g_{i}(τ_{i}): Average gain from processing a task of type i (points).

c_{i}: Average cost rate while processing a task of type i (points/second).

g_{i}(τ_{i}) − c_{i}τ_{i}: Average net gain from processing a task of type i (points).

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Random Variables

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Random Variables

g: Gross processing gain from k^{th} cycle.

c: Processing cost from k^{th} cycle.

τ: Processing time from k^{th} cycle.

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Random Variables

g: Gross processing gain from k^{th} cycle.

c: Processing cost from k^{th} cycle.

τ: Processing time from k^{th} cycle.

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Random Variables

g: Gross processing gain from k^{th} cycle.

c: Processing cost from k^{th} cycle.

τ: Processing time from k^{th} cycle.

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Random Variables

G̃_{k}: Total net gain from k^{th} cycle.

C̃_{k}: Total cost from k^{th} cycle.

T̃_{k}: Total time from k^{th} cycle.

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Random Processes

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Random Processes

G̃^{N} ≜ ∑_{k=1}^{N} G̃_{k}  C̃^{N} ≜ ∑_{k=1}^{N} C̃_{k}  T̃^{N} ≜ ∑_{k=1}^{N} T̃_{k}

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Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Statistics

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Statistics

\overline{g} ≜ E (g) = ∑_{i=1}^{n} \frac{λ_{i}}{λ} p_{i}g_{i}(τ_{i})

\overline{c} ≜ E (c) = ∑_{i=1}^{n} \frac{λ_{i}}{λ} p_{i}c_{i}τ_{i}

\overline{τ} ≜ E (τ) = ∑_{i=1}^{n} \frac{λ_{i}}{λ} p_{i}τ_{i}

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Statistics

var (g − c) = ∑_{i=1}^{n} \frac{λ_{i}}{λ} p_{i}(g_{i}(τ_{i}) − \overline{g})^{2}

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Statistics

E \binom{}{G^{N}} = N E \binom{}{G_{1}} = N (\overline{g} − \overline{c} − \frac{c^{s}}{λ} )
E \binom{}{C^{N}} = N E \binom{}{C_{1}} = N (\overline{c} + \frac{c^{s}}{λ} )
E \binom{}{T^{N}} = N E \binom{}{T_{1}} = N (\overline{τ} + \frac{1}{λ})

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Statistics

var \binom{}{G^{N}} = N var \binom{}{G_{1}} = N (var (\overline{g} − \overline{c}) + (\frac{c^{s}}{λ} )^{2})

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Search  Encounter  Process Ignore

Figure 1: Classical OFT Markov Renewal Process

Finite Lifetime?

Processing-Only Renewal Cycle

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Split Poisson Parameters

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Split Poisson Parameters

λ_{i}^{p}: Split Poisson encounter rate with processed tasks of type i (encounters/second).

λ^{p}: Merged Split Poisson encounter rate with all processed tasks (encounters/second).

λ_{i}^{p} ≜ p_{i}λ_{i}  λ^{p} ≜ ∑_{i=1}^{n} λ_{i}^{p}

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Split Poisson Parameters

λ_{i}^{p}: Split Poisson encounter rate with processed tasks of type i (encounters/second).

λ^{p}: Merged Split Poisson encounter rate with all processed tasks (encounters/second).

λ_{i}^{p} ≜ p_{i}λ_{i}  λ^{p} ≜ ∑_{i=1}^{n} λ_{i}^{p}

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Split Poisson Parameters

λ_{i}^{p}: Split Poisson encounter rate with processed tasks of type i (encounters/second).

λ^{p}: Merged Split Poisson encounter rate with all processed tasks (encounters/second).

λ_{i}^{p} ≜ p_{i}λ_{i}  λ^{p} ≜ ∑_{i=1}^{n} λ_{i}^{p}

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Split Poisson Parameters

λ_{i}^{p}: Split Poisson encounter rate with processed tasks of type i (encounters/second).

λ^{p}: Merged Split Poisson encounter rate with all processed tasks (encounters/second).

λ_{i}^{p} ≜ p_{i}λ_{i}  λ^{p} ≜ ∑_{i=1}^{n} λ_{i}^{p}

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Random Variables

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Random Variables

g^{p}: Gross processing gain from k^{th} processing cycle.

c^{p}: Processing cost from k^{th} processing cycle.

τ^{p}: Processing time from k^{th} processing cycle.

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Random Variables

G_{k}: Total net gain from k^{th} processing cycle.

C_{k}: Total cost from k^{th} processing cycle.

T_{k}: Total time from k^{th} processing cycle.

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Random Processes

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Random Processes

G^{N^{p}} ≜ ∑_{k=1}^{N^{p}} G_{k}  C^{N^{p}} ≜ ∑_{k=1}^{N^{p}} C_{k}  T^{N^{p}} ≜ ∑_{k=1}^{N^{p}} T_{k}

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Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Random Processes

G^{N^{p}} ≜ ∑_{k=1}^{N^{p}} G_{k}  C^{N^{p}} ≜ ∑_{k=1}^{N^{p}} C_{k}  T^{N^{p}} ≜ ∑_{k=1}^{N^{p}} T_{k}

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Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Statistics

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Statistics

\overline{g^{p}} ≜ E (g^{p}) = ∑_{i=1}^{n} \frac{λ_{i}^{p}}{λ^{p}} g_{i}(τ_{i})

\overline{c^{p}} ≜ E (c^{p}) = ∑_{i=1}^{n} \frac{λ_{i}^{p}}{λ^{p}} c_{i}τ_{i}

\overline{τ^{p}} ≜ E (τ^{p}) = ∑_{i=1}^{n} \frac{λ_{i}^{p}}{λ^{p}} τ_{i}

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Statistics

var (g^{p} − c^{p}) = ∑_{i=1}^{n} \frac{λ_{i}^{p}}{λ^{p}} (g_{i}(τ_{i}) − \overline{g^{p}})^{2}

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Statistics

E (G^{N^{p}}) = N^{p} E (G_{1}) = N^{p} (\overline{g^{p}} − \overline{c^{p}} − \frac{c^{s}}{λ^{p}} )
E (C^{N^{p}}) = N^{p} E (C_{1}) = N^{p} (\overline{c^{p}} + \frac{c^{s}}{λ^{p}} )
E (T^{N^{p}}) = N^{p} E (T_{1}) = N^{p} (\overline{τ^{p}} + \frac{1}{λ^{p}} )

Introduction Solitary Agent Model OFT Cycle Processing Cycle Optimization Results Remarks Questions?

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Find and Process

Figure 2: Finite Task Lifetime Markov Renewal Process

Statistics

var (G^{N^{p}}) = N^{p} var (G_{1}) = N^{p} (var (\overline{g^{p}} − \overline{c^{p}}) + (\frac{c^{s}}{λ^{p}} )^{2})

Optimization

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

Importance of Rate in Classical OFT

7 / 24 Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

8 / 24 Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

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when net gain is negative

Time minimizer

Net gain maximizer

Lifetime pressure

= =

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

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when net gain is negative

Time minimizer

Net gain maximizer

Lifetime pressure

= =

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

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when net gain is negative

Time minimizer

Net gain maximizer

Lifetime pressure

= =

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

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when net gain is negative

Time minimizer

Net gain maximizer

Lifetime pressure

aslim_{N→∞} \frac{G̃^{N}}{T̃^{N}} = \begin{matrix}E & ( & G̃_{1} & ) \\ E & ( & T̃_{1} & ) \end{matrix} = lim_{N→∞} E (\frac{G̃^{N}}{T̃^{N}} )

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

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when net gain is negative

Time minimizer

Net gain maximizer

Lifetime pressure

aslim_{N→∞} \frac{G̃^{N}}{T̃^{N}} = \begin{matrix}E & ( & G̃_{1} & ) \\ E & ( & T̃_{1} & ) \end{matrix} = lim_{N→∞} E (\frac{G̃^{N}}{T̃^{N}} )

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

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when net gain is negative

Time minimizer

Net gain maximizer

Lifetime pressure

aslim_{N→∞} \frac{G̃^{N}}{T̃^{N}} = \begin{matrix}E & ( & G̃_{1} & ) \\ E & ( & T̃_{1} & ) \end{matrix} = lim_{N→∞} E (\frac{G̃^{N}}{T̃^{N}} )

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

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when net gain is negative

Time minimizer

Net gain maximizer

Lifetime pressure

aslim_{N→∞} \frac{G̃^{N}}{T̃^{N}} = \begin{matrix}E & ( & G̃_{1} & ) \\ E & ( & T̃_{1} & ) \end{matrix} = lim_{N→∞} E (\frac{G̃^{N}}{T̃^{N}} )

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

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Time maximizer when net gain is negative

Time minimizer when net gain is positive

Net gain maximizer

Lifetime pressure

aslim_{N→∞} \frac{G̃^{N}}{T̃^{N}} = \begin{matrix}E & ( & G̃_{1} & ) \\ E & ( & T̃_{1} & ) \end{matrix} = lim_{N→∞} E (\frac{G̃^{N}}{T̃^{N}} )

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

8 / 24

Time maximizer when net gain is negative

Time minimizer when net gain is positive

Net gain maximizer

Lifetime pressure

aslim_{N→∞} \frac{G̃^{N}}{T̃^{N}} = \begin{matrix}E & ( & G̃_{1} & ) \\ E & ( & T̃_{1} & ) \end{matrix} = lim_{N→∞} E (\frac{G̃^{N}}{T̃^{N}} )

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

8 / 24

Time maximizer when net gain is negative

Time minimizer when net gain is positive

Net gain maximizer

Lifetime pressure

aslim_{N→∞} \frac{G̃^{N}}{T̃^{N}} = \begin{matrix}E & ( & G_{1} & ) \\ E & ( & T_{1} & ) \end{matrix} = lim_{N→∞} E (\frac{G̃^{N}}{T̃^{N}} )

Rate Maximization in Classical OFT

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\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

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\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{E(G̃_{1})}{E(T̃_{1})} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

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\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{E(G̃_{1})}{E(T̃_{1})} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

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\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{E(G̃_{1})}{E(T̃_{1})} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

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\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{g_{1}(τ_{1})  τ_{1} + \frac{1}{λ}}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

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\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{τ_{1} + \frac{1}{λ} g_{1}(τ_{1}) ^{\frac{1}{λ}}}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

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\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{τ_{1} + \frac{1}{λ} g_{1}(τ_{1}) ^{\frac{1}{λ}}}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

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\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix} ^{τ_{1} + \frac{1}{λ} g_{1}(τ_{1}) _{\frac{1}{λ}}  γ̃ g_{1}(τ_{1})  τ_{1}}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

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\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix} ^{τ_{1} + \frac{1}{λ} g_{1}(τ_{1}) _{\frac{1}{λ}}  γ̃ g_{1}(τ_{1})  τ_{1}}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

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\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix} ^{τ_{1} + \frac{1}{λ} g_{1}(τ_{1}) _{\frac{1}{λ}} g_{1}(τ̂_{1})  τ̂_{1} γ̃^{∗}}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

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\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix} ^{τ_{1} + \frac{1}{λ} g_{1}(τ_{1}) _{\frac{1}{λ}} g_{1}(τ̂_{1})  τ̂_{1} γ̃^{∗} g_{1}^{∗}}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

9 / 24

\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{τ_{1} g_{1}(τ_{1})  γ̃^{∗} g_{1}(τ̂_{1}) τ̂_{1}_{\frac{−1}{λ}}}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

9 / 24

\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix} ^{τ_{1} g_{1}(τ_{1})_{\frac{−1}{λ}}  γ̃^{∗} g_{1}(τ̂_{1}) τ̂_{1} \begin{matrix}MVT \\ g′_{1}(τ̂_{1}) = γ̃^{∗} \end{matrix}}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

9 / 24

\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{τ_{1} g_{1}(τ_{1}) λ < λ^{+}  ⋛^{MVT?}}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

9 / 24

\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{τ_{1} g_{1}(τ_{1})  γ̃^{∗} λ < λ^{+}_{\frac{−1}{λ^{+}}}}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

9 / 24

\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{τ_{1} g_{1}(τ_{1}) λ < λ^{+}_{\frac{−1}{λ^{+}}}  γ̃_{+}^{∗}}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

9 / 24

\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{τ_{1} g_{1}(τ_{1}) λ < λ^{+}_{\frac{−1}{λ^{+}}} τ̂_{1}^{+} g_{1}(τ̂_{1}^{+})  γ̃_{+}^{∗} γ̃ < γ̃_{+}^{∗}}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

9 / 24

\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix} ^{τ_{1} g_{1}(τ_{1}) λ^{−} < λ  ⋛^{MVT?}}

Figure 3: Visualization of Classical OFT Rate Maximization

Rate Maximization in Classical OFT

9 / 24

\begin{matrix}E(G̃_{1}) = g_{1}(τ_{1}) \\ E(T̃_{1}) = τ_{1} + \frac{1}{λ} \\ γ̃ ≜ \frac{g_{1}(τ_{1})}{τ_{1}+ \frac{1}{λ}} \\ λ = λ_{1}, p_{1} = 1 \\ ♢^{∗} ≜ max \{♢\} \end{matrix} ^{τ_{1} g_{1}(τ_{1})_{\frac{−1}{λ^{−}}}  τ̂_{1}^{−} g_{1}(τ̂_{1}^{−})  γ̃_{−}^{∗} λ^{−} < λ  γ̃_{−}^{∗} < γ̃}

Figure 3: Visualization of Classical OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{E(G̃_{1})}{E(T̃_{1})} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} \frac{λ_{i}}{λ} p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} \frac{λ_{i}}{λ} p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} \frac{λ_{i}}{λ} p_{i}c_{i}τ_{i}

\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{E(G̃_{1})}{E(T̃_{1})} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} \frac{λ_{i}}{λ} p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} \frac{λ_{i}}{λ} p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} \frac{λ_{i}}{λ} p_{i}c_{i}τ_{i}

\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{\overline{g}−\overline{c}− \frac{c^{s}}{λ}}{\overline{τ}+ \frac{1}{λ}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} \frac{λ_{i}}{λ} p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} \frac{λ_{i}}{λ} p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} \frac{λ_{i}}{λ} p_{i}c_{i}τ_{i}

\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{\overline{g}−\overline{c}− \frac{c^{s}}{λ}}{\overline{τ}+ \frac{1}{λ}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}c_{i}τ_{i}

\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{\overline{g}−\overline{c}− \frac{c^{s}}{λ}}{\overline{τ}+ \frac{1}{λ}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}c_{i}τ_{i}
\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{\overline{g}−\overline{c}− \frac{c^{s}}{λ}}{\overline{τ}+ \frac{1}{λ}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{\overline{g} − \overline{c} − \frac{c^{s}}{λ}  \overline{τ} + \frac{1}{λ}}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}c_{i}τ_{i}
\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{\overline{g}−\overline{c}− \frac{c^{s}}{λ}}{\overline{τ}+ \frac{1}{λ}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{\overline{g} − \overline{c} − \frac{c^{s}}{λ}  \overline{τ} + \frac{1}{λ}}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}c_{i}τ_{i}
\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{\overline{g}−\overline{c}− \frac{c^{s}}{λ}}{\overline{τ}+ \frac{1}{λ}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{\overline{g} − \overline{c} − \frac{c^{s}}{λ}  \overline{τ} + \frac{1}{λ}}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}c_{i}τ_{i}
\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{\overline{g}−\overline{c}− \frac{c^{s}}{λ}}{\overline{τ}+ \frac{1}{λ}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{\overline{g} − \overline{c} − \frac{c^{s}}{λ}  \overline{τ} + \frac{1}{λ}}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}c_{i}τ_{i}
\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{\overline{g}−\overline{c}− \frac{c^{s}}{λ}}{\overline{τ}+ \frac{1}{λ}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{\overline{g} − \overline{c} − \frac{c^{s}}{λ}  \overline{τ} + \frac{1}{λ}}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}c_{i}τ_{i}
\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{\overline{g}−\overline{c}− \frac{c^{s}}{λ}}{\overline{τ}+ \frac{1}{λ}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{\overline{τ} + \frac{1}{λ} \overline{g} − \overline{c} − \frac{c^{s}}{λ}^{\frac{1}{λ}}^{\frac{−c^{s}}{λ}}}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}c_{i}τ_{i}
\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{\overline{g}−\overline{c}− \frac{c^{s}}{λ}}{\overline{τ}+ \frac{1}{λ}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{\overline{τ} + \frac{1}{λ} \overline{g} − \overline{c} − \frac{c^{s}}{λ}^{\frac{1}{λ}}^{\frac{−c^{s}}{λ}}}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}c_{i}τ_{i}
\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{\overline{g}−\overline{c}− \frac{c^{s}}{λ}}{\overline{τ}+ \frac{1}{λ}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{\overline{g} − \overline{c}  \overline{τ}}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}c_{i}τ_{i}
\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{\overline{g}−\overline{c}− \frac{c^{s}}{λ}}{\overline{τ}+ \frac{1}{λ}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{\overline{τ} \overline{g} − \overline{c}_{\frac{−1}{λ}} _{\frac{c^{s}}{λ}} −c^{s}  γ̃^{∗}}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}c_{i}τ_{i}
\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{\overline{g}−\overline{c}− \frac{c^{s}}{λ}}{\overline{τ}+ \frac{1}{λ}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{\overline{τ} \overline{g} − \overline{c}_{\frac{−1}{λ}} _{\frac{c^{s}}{λ}} −c^{s}}

Figure 4: Visualization of Generalized OFT Rate Maximization

Efficient OFT Rate Maximization

10 / 24

\overline{τ} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}τ_{i}  \overline{g} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}g_{i}(τ_{i})  \overline{c} ≜ ∑_{i=1}^{n} Λ_{i}p_{i}c_{i}τ_{i}
\begin{matrix}E(G̃_{1}) = \overline{g} − \overline{c} − \frac{c^{s}}{λ} \\ E(T̃_{1}) = \overline{τ} + \frac{1}{λ} \\ γ̃ ≜ \frac{\overline{g}−\overline{c}− \frac{c^{s}}{λ}}{\overline{τ}+ \frac{1}{λ}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}  ^{\overline{τ} \overline{g} − \overline{c}_{\frac{−1}{λ}} _{\frac{c^{s}}{λ}} −c^{s}  γ̃^{∗} = 0}

Figure 4: Visualization of Generalized OFT Rate Maximization

Finite-Lifetime Approach

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

11 / 24

Lifetime ends after N^{p} processed tasks

Lifetime success threshold of G^{T} points

Average cycle success threshold of \frac{G^{T}}{N^{p}} points

Blends rate maximization and risk sensitivity

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

11 / 24

Lifetime ends after N^{p} processed tasks

Lifetime success threshold of G^{T} points

Average cycle success threshold of \frac{G^{T}}{N^{p}} points

Blends rate maximization and risk sensitivity

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

11 / 24

Lifetime ends after N^{p} processed tasks

Lifetime success threshold of G^{T} points

Average cycle success threshold of \frac{G^{T}}{N^{p}} points

Blends rate maximization and risk sensitivity

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

11 / 24

Lifetime ends after N^{p} processed tasks

Lifetime success threshold of G^{T} points

Average cycle success threshold of \frac{G^{T}}{N^{p}} points

Blends rate maximization and risk sensitivity

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

11 / 24

Lifetime ends after N^{p} processed tasks

Lifetime success threshold of G^{T} points

Average cycle success threshold of \frac{G^{T}}{N^{p}} points

Blends rate maximization and risk sensitivity

Finite-Lifetime Rate Maximization

12 / 24

\begin{matrix}μ ≜ E(G_{1}) = \overline{g^{p}} − \overline{c^{p}} − \frac{c^{s}}{λ^{p}} \\ E(T_{1}) = \overline{τ^{p}} + \frac{1}{λ^{p}} \\ γ ≜ \frac{μ− \frac{G^{T}}{N^{p}}}{\overline{τ^{p}}+ \frac{1}{λ^{p}}} \\ ♢^{∗} ≜ max \{♢\} \end{matrix}^{\overline{τ^{p}} \overline{g^{p}} − \overline{c^{p}}  γ^{∗}_{\frac{G^{T}}{N^{p}}} \frac{G^{T}}{N^{p}} + \frac{c^{s}}{λ^{p}} −c^{s}_{(\overline{g^{p}} − \overline{c^{p}})^{∗}}  _{(\overline{τ^{p}})^{∗}}_{\frac{−1}{λ^{p}}}}

Figure 5: Excess Rate Maximization (N^{p} cycles and G^{T} threshold)

Reward-to-Variability Maximization

13 / 24

μ ≜ E(G_{1}) = \overline{g^{p}} − \overline{c^{p}} − \frac{c^{s}}{λ^{p}} σ ≜ std(G_{1}) = \sqrt{var (g^{p} − c^{p}) + (\frac{c^{s}}{λ^{p}} )^{2}}

ρ ≜ ^{\frac{μ− \frac{G^{T}}{N^{p}}}{σ}} ♢^{∗} ≜ max \{♢\}

σ − \frac{|c^{s}|}{λ^{p}}

\overline{g^{p}} − \overline{c^{p}}
ρ^{∗}

− sgn(c^{s})

\frac{G^{T}}{N^{p}}

−\frac{|c^{s}|}{λ^{p}} σ^{∗} − \frac{|c^{s}|}{λ^{p}}

(\overline{g^{p}} − \overline{c^{p}})^{∗}

Figure 6: Sharpe Ratio Maximization (N^{p} cycles and G^{T} threshold)

Optimization by Discounting

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

14 / 24 Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

14 / 24

Objectives A and B

Discount w ∈ ℝ_{≥0}

Maximization of A − wB is Pareto optimal

Opportunity cost

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

14 / 24

Objectives A and B

Discount w ∈ ℝ_{≥0}

Maximization of A − wB is Pareto optimal

Opportunity cost

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

14 / 24

Objectives A and B

Discount w ∈ ℝ_{≥0}

Maximization of A − wB is Pareto optimal

Opportunity cost

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

14 / 24

Objectives A and B

Discount w ∈ ℝ_{≥0}

Maximization of A − wB is Pareto optimal

Opportunity cost

E \binom{}{G_{1}} − E \binom{}{T_{1}}

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

14 / 24

Objectives A and B

Discount w ∈ ℝ_{≥0}

Maximization of A − wB is Pareto optimal

Opportunity cost

E \binom{}{G_{1}} − E \binom{}{T_{1}}

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

14 / 24

Objectives A and B

Discount w ∈ ℝ_{≥0}

Maximization of A − wB is Pareto optimal

Opportunity cost

E \binom{}{G_{1}} − E \binom{}{T_{1}}

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

14 / 24

Objectives A and B

Discount w ∈ ℝ_{≥0}

Maximization of A − wB is Pareto optimal

Opportunity cost

E \binom{}{G_{1}} − wE \binom{}{T_{1}}

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

14 / 24

Objectives A and B

Discount γ̃^{∗} ∈ ℝ

Maximization of A − γ̃^{∗}B is Pareto optimal

Opportunity cost

E \binom{}{G_{1}} − γ̃^{∗}E \binom{}{T_{1}}

Introduction

Solitary Agent Model

Optimization OFT Rate OFT Optimization OFT & Pareto Finite Approach Finite Rate Finite R-to-V Pareto Discounts

Results

Remarks

Questions?

14 / 24

Other Possibilities

E \binom{}{G_{1}} − w̃_{0} E \binom{}{T_{1}}

E \binom{}{G_{1}} − w̃_{1} std \binom{}{G_{1}}

E (G̃_{1} + C̃_{1}) − w̃_{2} E \binom{}{C_{1}}

E (G_{1}) − w_{0} E (T_{1})

E (G_{1}) − w_{1} std (G_{1})

E (G_{1} + C_{1}) − w_{2} E (C_{1})

Results

Introduction Solitary Agent Model Optimization Results A-to-D Functions Lumped Tasks Variable Time Remarks Questions?

Advantage-to-Disadvantage Ratio

15 / 24 Introduction Solitary Agent Model Optimization Results A-to-D Functions Lumped Tasks Variable Time Remarks Questions?

16 / 24 Introduction Solitary Agent Model Optimization Results A-to-D Functions Lumped Tasks Variable Time Remarks Questions?

16 / 24

\begin{aligned}A & ≜ a + ∑_{i=1}^{n} p_{i}a_{i}(τ_{i}) \\ D & ≜ d + ∑_{i=1}^{n} p_{i}d_{i}(τ_{i}) \\ & J ≜ \frac{A}{D} \end{aligned}

Introduction Solitary Agent Model Optimization Results A-to-D Functions Lumped Tasks Variable Time Remarks Questions?

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\begin{aligned}A & ≜ a + ∑_{i=1}^{n} p_{i}a_{i}(τ_{i}) \\ D & ≜ d + ∑_{i=1}^{n} p_{i}d_{i}(τ_{i}) \\ & J ≜ \frac{A}{D} \end{aligned}

Introduction Solitary Agent Model Optimization Results A-to-D Functions Lumped Tasks Variable Time Remarks Questions?

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\begin{aligned}A & ≜ a + ∑_{i=1}^{n} p_{i}a_{i}(τ_{i}) \\ D & ≜ d + ∑_{i=1}^{n} p_{i}d_{i}(τ_{i}) \\ & J ≜ \frac{A}{D} \end{aligned}

Introduction Solitary Agent Model Optimization Results A-to-D Functions Lumped Tasks Variable Time Remarks Questions?

Examples: Gain and Time Objectives

\frac{E(G_{1}) − \frac{G^{T}}{N^{p}}}{E(T_{1})} = \frac{−c^{s} + ∑_{i=1}^{n} p_{i}λ_{i} (g_{i}(τ_{i}) − c_{i}τ_{i} − \frac{G^{T}}{N^{p}} )}{1 + ∑_{i=1}^{n} p_{i}λ_{i}τ_{i}}

E(G_{1})−w E(T_{1}) = \frac{−(c^{s} + w) + ∑_{i=1}^{n} p_{i}λ_{i} (g_{i}(τ_{i}) − c_{i}τ_{i} − w_{i}τ_{i})}{∑_{i=1}^{n} p_{i}λ_{i}}

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Introduction Solitary Agent Model Optimization Results A-to-D Functions Lumped Tasks Variable Time Remarks Questions?

Examples: Efficiency-Type Objectives

\frac{E(G_{1}) + E(C_{1}) − \frac{G_{g}^{T}}{N^{p}}}{E(C_{1})} = \frac{∑_{i=1}^{n} p_{i}λ_{i} (g_{i}(τ_{i}) − \frac{G_{g}^{T}}{N^{p}} )}{c^{s} + ∑_{i=1}^{n} p_{i}λ_{i}c_{i}τ_{i}}

E(G_{1})+E(C_{1})−w E(C_{1}) = \frac{−wc^{s} + ∑_{i=1}^{n} p_{i}λ_{i} (g_{i}(τ_{i}) − w_{i}c_{i}τ_{i})}{∑_{i=1}^{n} p_{i}λ_{i}}

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Constant Disadvantage Case

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Assume that

d_{j} is constant for all τ_{j}

\frac{a_{j}}{d_{j}} achieves its maximum at τ_{j}^{∗}

Index so that

\frac{a_{1}(τ_{1}^{∗})}{d_{1}(τ_{1}^{∗})} > \frac{a_{2}(τ_{2}^{∗})}{d_{2}(τ_{2}^{∗})} > ⋯ > \frac{a_{n−1}(τ_{n−1}^{∗})}{d_{n−1}(τ_{n−1}^{∗})} > \frac{a_{n}(τ_{n}^{∗})}{d_{n}(τ_{n}^{∗})}

Exclude when

\frac{a + ∑_{i=1}^{k} a_{i}(τ_{i}^{∗})}{d + ∑_{i=1}^{k} d_{i}(τ_{i}^{∗})} > \frac{a_{k+1}(τ_{k+1}^{∗})}{d_{k+1}(τ_{k+1}^{∗})}

Introduction Solitary Agent Model Optimization Results A-to-D Functions Lumped Tasks Variable Time Remarks Questions?

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Assume that

d_{j} is constant for all τ_{j}

\frac{a_{j}}{d_{j}} achieves its maximum at τ_{j}^{∗}

Index so that

\frac{a_{1}(τ_{1}^{∗})}{d_{1}(τ_{1}^{∗})} > \frac{a_{2}(τ_{2}^{∗})}{d_{2}(τ_{2}^{∗})} > ⋯ > \frac{a_{n−1}(τ_{n−1}^{∗})}{d_{n−1}(τ_{n−1}^{∗})} > \frac{a_{n}(τ_{n}^{∗})}{d_{n}(τ_{n}^{∗})}

Exclude when

\frac{a + ∑_{i=1}^{k} a_{i}(τ_{i}^{∗})}{d + ∑_{i=1}^{k} d_{i}(τ_{i}^{∗})} > \frac{a_{k+1}(τ_{k+1}^{∗})}{d_{k+1}(τ_{k+1}^{∗})}

Introduction Solitary Agent Model Optimization Results A-to-D Functions Lumped Tasks Variable Time Remarks Questions?

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Assume that

d_{j} is constant for all τ_{j}

\frac{a_{j}}{d_{j}} achieves its maximum at τ_{j}^{∗}

Index so that

\frac{a_{1}(τ_{1}^{∗})}{d_{1}(τ_{1}^{∗})} > \frac{a_{2}(τ_{2}^{∗})}{d_{2}(τ_{2}^{∗})} > ⋯ > \frac{a_{n−1}(τ_{n−1}^{∗})}{d_{n−1}(τ_{n−1}^{∗})} > \frac{a_{n}(τ_{n}^{∗})}{d_{n}(τ_{n}^{∗})}

Exclude when

\frac{a + ∑_{i=1}^{k} a_{i}(τ_{i}^{∗})}{d + ∑_{i=1}^{k} d_{i}(τ_{i}^{∗})} > \frac{a_{k+1}(τ_{k+1}^{∗})}{d_{k+1}(τ_{k+1}^{∗})}

Generalized MVT

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Assume that

(\frac{a_{j}}{d_{j}} )′ < 0 for allτ_{j} (maximum atτ_{j}^{∗} = 0)

Index so that

\frac{a_{1}(0)}{d_{1}(0)} > \frac{a_{2}(0)}{d_{2}(0)} > ⋯ > \frac{a_{n−1}(0)}{d_{n−1}(0)} > \frac{a_{n}(0)}{d_{n}(0)}

Exclude when

\frac{a + ∑_{i=1}^{k} a_{i}(τ_{i}^{k})}{d + ∑_{i=1}^{k} d_{i}(τ_{i}^{k})} > \frac{a_{k+1}(0)}{d_{k+1}(0)}

Introduction Solitary Agent Model Optimization Results A-to-D Functions Lumped Tasks Variable Time Remarks Questions?

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Assume that

(\frac{a_{j}}{d_{j}} )′ < 0 for allτ_{j} (maximum atτ_{j}^{∗} = 0)

Index so that

\frac{a_{1}(0)}{d_{1}(0)} > \frac{a_{2}(0)}{d_{2}(0)} > ⋯ > \frac{a_{n−1}(0)}{d_{n−1}(0)} > \frac{a_{n}(0)}{d_{n}(0)}

Exclude when

\frac{a + ∑_{i=1}^{k} a_{i}(τ_{i}^{k})}{d + ∑_{i=1}^{k} d_{i}(τ_{i}^{k})} > \frac{a_{k+1}(0)}{d_{k+1}(0)}

Introduction Solitary Agent Model Optimization Results A-to-D Functions Lumped Tasks Variable Time Remarks Questions?

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Assume that

(\frac{a_{j}}{d_{j}} )′ < 0 for allτ_{j} (maximum atτ_{j}^{∗} = 0)

Index so that

\frac{a_{1}(0)}{d_{1}(0)} > \frac{a_{2}(0)}{d_{2}(0)} > ⋯ > \frac{a_{n−1}(0)}{d_{n−1}(0)} > \frac{a_{n}(0)}{d_{n}(0)}

Exclude when

\frac{a + ∑_{i=1}^{k} a_{i}(τ_{i}^{k})}{d + ∑_{i=1}^{k} d_{i}(τ_{i}^{k})} > \frac{a_{k+1}(0)}{d_{k+1}(0)}

Introduction Solitary Agent Model Optimization Results A-to-D Functions Lumped Tasks Variable Time Remarks Questions?

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Assume that

(\frac{a_{j}}{d_{j}} )′ < 0 for allτ_{j} (maximum atτ_{j}^{∗} = 0)

Index so that

\frac{a_{1}(0)}{d_{1}(0)} > \frac{a_{2}(0)}{d_{2}(0)} > ⋯ > \frac{a_{n−1}(0)}{d_{n−1}(0)} > \frac{a_{n}(0)}{d_{n}(0)}

Exclude when

\frac{a + ∑_{i=1}^{k} a_{i}(τ_{i}^{k})}{d + ∑_{i=1}^{k} d_{i}(τ_{i}^{k})} > \frac{a_{k+1}(0)}{d_{k+1}(0)}

Introduction Solitary Agent Model Optimization Results A-to-D Functions Lumped Tasks Variable Time Remarks Questions?

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where τ_{j}^{k} is found with

\frac{a′_{k}(τ_{j}^{k})}{d′_{k}(τ_{j}^{k})} = \frac{a + ∑_{i=1}^{k} a_{i}(τ_{i}^{k})}{d + ∑_{i=1}^{k} d_{i}(τ_{i}^{k})}

This is a generalized version of the marginal value theorem.

Introduction Solitary Agent Model Optimization Results A-to-D Functions Lumped Tasks Variable Time Remarks Questions?

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where τ_{j}^{k} is found with

\frac{a′_{k}(τ_{j}^{k})}{d′_{k}(τ_{j}^{k})} = \frac{a + ∑_{i=1}^{k} a_{i}(τ_{i}^{k})}{d + ∑_{i=1}^{k} d_{i}(τ_{i}^{k})}

This is a generalized version of the marginal value theorem.

Concluding Remarks

Introduction Solitary Agent Model Optimization Results Remarks Questions?

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Simple stochastic model of a solitary agent

Provided analytical tools for finding optimal agent behavior

Model can be expanded (e.g., recognition cost, behavior-dependent rates, nonlinear fuel costs)

Optimization approaches can be enhanced using methods from post-modern portfolio theory

Behaviors should be implemented on real agents

Introduction Solitary Agent Model Optimization Results Remarks Questions?

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Simple stochastic model of a solitary agent

Provided analytical tools for finding optimal agent behavior

Model can be expanded (e.g., recognition cost, behavior-dependent rates, nonlinear fuel costs)

Optimization approaches can be enhanced using methods from post-modern portfolio theory

Behaviors should be implemented on real agents

Introduction Solitary Agent Model Optimization Results Remarks Questions?

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Simple stochastic model of a solitary agent

Provided analytical tools for finding optimal agent behavior

Model can be expanded (e.g., recognition cost, behavior-dependent rates, nonlinear fuel costs)

Optimization approaches can be enhanced using methods from post-modern portfolio theory

Behaviors should be implemented on real agents

Introduction Solitary Agent Model Optimization Results Remarks Questions?

19 / 24

Simple stochastic model of a solitary agent

Provided analytical tools for finding optimal agent behavior

Model can be expanded (e.g., recognition cost, behavior-dependent rates, nonlinear fuel costs)

Optimization approaches can be enhanced using methods from post-modern portfolio theory

Behaviors should be implemented on real agents

Introduction Solitary Agent Model Optimization Results Remarks Questions?

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Simple stochastic model of a solitary agent

Provided analytical tools for finding optimal agent behavior

Model can be expanded (e.g., recognition cost, behavior-dependent rates, nonlinear fuel costs)

Optimization approaches can be enhanced using methods from post-modern portfolio theory

Behaviors should be implemented on real agents

Introduction Solitary Agent Model Optimization Results Remarks Questions?

19 / 24

Simple stochastic model of a solitary agent

Provided analytical tools for finding optimal agent behavior

Model can be expanded (e.g., recognition cost, behavior-dependent rates, nonlinear fuel costs)

Optimization approaches can be enhanced using methods from post-modern portfolio theory

Behaviors should be implemented on real agents

Questions?

Introduction Solitary Agent Model Optimization Results Remarks Questions?

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Major Influences

E.L. Charnov. Optimal foraging: The marginal value theorem. Theoretical Population Biology, 9(2): 129–136, April 1976.

Eric L. Charnov and Gordon H. Orians. Optimal foraging: some theoretical explorations. PhD thesis, University of Washington, 1973.

D.W. Stephens and J.R. Krebs. Foraging Theory. Princeton Univ. Press, Princeton, NJ, 1986.

Follow-Up

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

Applications

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22 / 24 Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Military

Surveillance

Offensive action

Robotic Exploration (e.g., DEPTHX)

Deep Sea

Space

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Military

Surveillance

Offensive action

Robotic Exploration (e.g., DEPTHX)

Deep Sea

Space

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Limited resources

Retrieval space

Objects to deploy

Prioritization during real-time search

Failure thresholds (e.g., G^{T})

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Limited resources

Retrieval space

Objects to deploy

Prioritization during real-time search

Failure thresholds (e.g., G^{T})

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Limited resources

Retrieval space

Objects to deploy

Prioritization during real-time search

Failure thresholds (e.g., G^{T})

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Limited resources

Retrieval space

Objects to deploy

Prioritization during real-time search

Failure thresholds (e.g., G^{T})

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Information foraging of human beings (i.e., analysis)

Web behavior consistent with foraging predictions

Design technology based on behavioral analysis

Content delivery based on encounter characteristics of search

Advertising

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Information foraging of human beings (i.e., analysis)

Web behavior consistent with foraging predictions

Design technology based on behavioral analysis

Content delivery based on encounter characteristics of search

Advertising

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Information foraging of human beings (i.e., analysis)

Web behavior consistent with foraging predictions

Design technology based on behavioral analysis

Content delivery based on encounter characteristics of search

Advertising

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Information foraging of human beings (i.e., analysis)

Web behavior consistent with foraging predictions

Design technology based on behavioral analysis

Content delivery based on encounter characteristics of search

Advertising

Future Directions

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Model can be expanded

Behavior-dependent encounter rates

Recognition costs

Nonlinear cost functions

Parameter uncertainty and estimation

Relax Poisson assumption

New stochastic optimization criteria

Skew-sensitive objectives

Use of lower-partial moments

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Model can be expanded

Behavior-dependent encounter rates

Recognition costs

Nonlinear cost functions

Parameter uncertainty and estimation

Relax Poisson assumption

New stochastic optimization criteria

Skew-sensitive objectives

Use of lower-partial moments

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Model can be expanded

Behavior-dependent encounter rates

Recognition costs

Nonlinear cost functions

Parameter uncertainty and estimation

Relax Poisson assumption

New stochastic optimization criteria

Skew-sensitive objectives

Use of lower-partial moments

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Model can be expanded

Behavior-dependent encounter rates

Recognition costs

Nonlinear cost functions

Parameter uncertainty and estimation

Relax Poisson assumption

New stochastic optimization criteria

Skew-sensitive objectives

Use of lower-partial moments

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

23 / 24

Model can be expanded

Behavior-dependent encounter rates

Recognition costs

Nonlinear cost functions

Parameter uncertainty and estimation

Relax Poisson assumption

New stochastic optimization criteria

Skew-sensitive objectives

Use of lower-partial moments

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Model can be expanded

Behavior-dependent encounter rates

Recognition costs

Nonlinear cost functions

Parameter uncertainty and estimation

Relax Poisson assumption

New stochastic optimization criteria

Skew-sensitive objectives

Use of lower-partial moments

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

23 / 24

Model can be expanded

Behavior-dependent encounter rates

Recognition costs

Nonlinear cost functions

Parameter uncertainty and estimation

Relax Poisson assumption

New stochastic optimization criteria

Skew-sensitive objectives

Use of lower-partial moments

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

23 / 24

Model can be expanded

Behavior-dependent encounter rates

Recognition costs

Nonlinear cost functions

Parameter uncertainty and estimation

Relax Poisson assumption

New stochastic optimization criteria

Skew-sensitive objectives

Use of lower-partial moments

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Model can be expanded

Behavior-dependent encounter rates

Recognition costs

Nonlinear cost functions

Parameter uncertainty and estimation

Relax Poisson assumption

New stochastic optimization criteria

Skew-sensitive objectives

Use of lower-partial moments

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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PMPT: Stochastic dominance

Static approach

Embraces risk sensitivity and a wider range of return distributions

State-based dynamic programming methods

Modern approach

Improves performance

Increases behavioral complexity

Better machines and human-in-loop systems prevent complexity problems

Parameter estimation can be built-in

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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PMPT: Stochastic dominance

Static approach

Embraces risk sensitivity and a wider range of return distributions

State-based dynamic programming methods

Modern approach

Improves performance

Increases behavioral complexity

Better machines and human-in-loop systems prevent complexity problems

Parameter estimation can be built-in

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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PMPT: Stochastic dominance

Static approach

Embraces risk sensitivity and a wider range of return distributions

State-based dynamic programming methods

Modern approach

Improves performance

Increases behavioral complexity

Better machines and human-in-loop systems prevent complexity problems

Parameter estimation can be built-in

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

23 / 24

PMPT: Stochastic dominance

Static approach

Embraces risk sensitivity and a wider range of return distributions

State-based dynamic programming methods

Modern approach

Improves performance

Increases behavioral complexity

Better machines and human-in-loop systems prevent complexity problems

Parameter estimation can be built-in

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

23 / 24

PMPT: Stochastic dominance

Static approach

Embraces risk sensitivity and a wider range of return distributions

State-based dynamic programming methods

Modern approach

Improves performance

Increases behavioral complexity

Better machines and human-in-loop systems prevent complexity problems

Parameter estimation can be built-in

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

23 / 24

PMPT: Stochastic dominance

Static approach

Embraces risk sensitivity and a wider range of return distributions

State-based dynamic programming methods

Modern approach

Improves performance

Increases behavioral complexity

Better machines and human-in-loop systems prevent complexity problems

Parameter estimation can be built-in

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

23 / 24

PMPT: Stochastic dominance

Static approach

Embraces risk sensitivity and a wider range of return distributions

State-based dynamic programming methods

Modern approach

Improves performance

Increases behavioral complexity

Better machines and human-in-loop systems prevent complexity problems

Parameter estimation can be built-in

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

23 / 24

PMPT: Stochastic dominance

Static approach

Embraces risk sensitivity and a wider range of return distributions

State-based dynamic programming methods

Modern approach

Improves performance

Increases behavioral complexity

Better machines and human-in-loop systems prevent complexity problems

Parameter estimation can be built-in

Questions?

Introduction Solitary Agent Model Optimization Results Remarks Questions? Follow-Up Applications Future Directions Questions?

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Influences for Future Work

V.S. Bawa. Optimal rules for ordering uncertain prospects. Journal of Financial Economics, 2:95–121, 1975.

Vijay S. Bawa. Stochastic dominance: A research bibliography. Management Science, 28(6):698–712, June 1982.

Vijay S. Bawa. Safety-first, stochastic dominance, and optimal portfolio choice. The Journal of Financial and Quantitative Analysis, 13(2):255–271, June 1978.

Vijay S. Bawa and Eric B. Lindenberg. Capital market equilibrium in a mean-lower partial moment framework. Journal of Financial Economics, 5:189–200, 1977.

Thomas Caraco. On foraging time allocation in a stochastic environment. Ecology, 61(1): 119–128, 1980.

A. Houston and J. McNamara. Models of Adaptive Behavior. Cambridge Univ. Press, Cambridge, 1999.

Marc Mangel and Colin W. Clark. Dynamic Modeling in Behavioral Ecology. Princeton University Press, New Jersey, 1988.