Overview and examples of implementation and application of Kelly criteria and fairness-aware optimisation algorithms.

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On the Kelly criteria and optimisation algorithms for fairness

Methods used for various capital allocations include the Kelly criterion and fairness-aware optimisation algorithms.

The Kelly criterion is a method for optimising the allocation of capital in gambling and investment, which calculates how much money should be invested when the expected value of an investment or bet is positive. The Kelly criterion is a measure that enables maximising long-term asset growth.

The Kelly criterion is expressed by the following formula

\[
f^* = \frac{bp – q}{b}
\]

Where,
– \( f^* \) is the proportion of capital to be invested,
– \( b \) is the odds (return) of the bet,
– \( p \) is the probability of winning,
– \( q \) is the probability of losing （\( q = 1 – p \)）.

Applications in which the Kelly criterion is used include money management, optimising investment portfolios and designing gambling strategies.

Fairness-aware optimisation aims to maintain equal benefits and satisfaction for each agent when allocating a particular resource to multiple agents (individuals, groups, systems, etc.) and is used in pricing and resource allocation optimisation.

Examples of algorithms used in fairness-aware optimisation include: determining the amount of resources to allocate to each agent using several algorithms (e.g. minimum fairness, maximum minimum fairness), ‘fair resource allocation’ to ensure minimum satisfaction, each agent acting cooperatively or competitively Scenarios are modelled and fairness-aware strategies are designed. Examples include ‘applying game theory’, which uses Nash equilibrium to aim for a state where each agent’s strategy is mutually optimal, and multi-objective optimisation, which considers multiple objectives simultaneously, balancing agent satisfaction and the efficient allocation of resources, analysing trade-offs and finding compromises.

The Kelly criterion is a powerful method for providing an optimal allocation of capital, and fairness-aware optimisation algorithms are important for achieving an equitable allocation of resources, especially in systems involving multiple agents, which, when combined, can lead to more sustainable and equitable systems This can be achieved by combining them.

implementation example

This section describes a simple example of using Python to calculate the Kelly criterion and an example of resource allocation with equity considerations.

1. example implementation of the Kelly criterion: the following Python code calculates the proportion of capital to be wagered based on the Kelly criterion.

def kelly_criterion(prob_win, odds):
    """Functions to calculate the Kelly criterion."""
    prob_loss = 1 - prob_win
    kelly_fraction = (odds * prob_win - prob_loss) / odds
    return kelly_fraction

# As an example, if the winning percentage is 60% and the odds are 2.0
prob_win = 0.6
odds = 2.0
bet_fraction = kelly_criterion(prob_win, odds)

print(f"Percentage of capital to be invested: {bet_fraction:.2f}")

2. example implementation of fairness-aware resource allocation: next, as an example of simple fairness-aware resource allocation, Python code is shown that equally allocates available resources to multiple agents.

def allocate_resources(total_resources, num_agents):
    """Functions to distribute resources equally to agents."""
    allocation = [total_resources // num_agents] * num_agents
    remainder = total_resources % num_agents
    
    # The remaining resources are allocated on a first-come, first-served basis.
    for i in range(remainder):
        allocation[i] += 1
        
    return allocation

# If total resources are 100 and the number of agents is 4
total_resources = 100
num_agents = 4
allocation = allocate_resources(total_resources, num_agents)

print(f"Resource allocation to agents: {allocation}")

3. example implementation of a fairness-aware optimisation algorithm: an example of minimum fairness-aware resource allocation is presented. Here, a simple algorithm is implemented to allocate resources based on agent satisfaction.

import numpy as np

def fair_allocation(total_resources, agent_needs):
    """Resource allocation function with minimum fairness."""
    num_agents = len(agent_needs)
    allocation = np.zeros(num_agents)

    # Minimum needs to be met.
    min_need = min(agent_needs)
    
    # Allocate minimum needs to each agent.
    for i in range(num_agents):
        allocation[i] = min(min_need, total_resources)
        total_resources -= allocation[i]
    
    # Equal distribution of remaining resources
    if total_resources > 0:
        allocation += total_resources / num_agents

    return allocation

# If total resources are 100 and agent needs are [20, 50, 30, 40].
total_resources = 100
agent_needs = [20, 50, 30, 40]
allocation = fair_allocation(total_resources, agent_needs)

print(f"Equitable resource allocation: {allocation}")

The execution results are as follows.

1. Kerry Criteria Execution result:
   • Ratio of capital to be invested: 0.20 (e.g.)
2. Resource allocation run result:
   • Resource allocation to agents: [25, 25, 25, 25] (example)
3. Result of resource allocation run with minimum fairness:
   • Fair resource allocation: [20, 20, 20, 20, 40] (example)

Application examples

Examples of applications of the Kelly Criterion and fairness-aware optimisation algorithms include the following.

1. gambling and investing: gamblers and investors can use the Kelly criterion to determine how much money to allocate to a particular bet or investment. This enables them to maximise long-term asset growth. This can be used, for example, in horse race betting or equity investment, where the probability of winning and return are taken into account when determining the appropriate stake.

2. online platform pricing: surge pricing described in”Machine learning and algorithms used for surge pricing and examples of implementation” is used in the travel and hospitality industry. In this context, it is important to set appropriate prices for different users based on income and geography, with a view to fairness. For example, use scenarios could include offering special discounts to some customers when demand increases during certain periods, in order to maintain overall satisfaction.

3. allocation of healthcare resources: healthcare systems need to ensure that limited resources (e.g. healthcare workers and beds) are distributed equitably among patients. For this purpose, the use of equity-aware algorithms has been implemented to allocate healthcare services appropriately according to the severity and needs of patients.

4. allocation of educational resources: minimum equity-aware algorithms are used in educational institutions to ensure equitable distribution of educational resources (e.g. teachers, teaching materials, facilities) to students and classes. For example, priorities are set to meet the needs of specific regions or schools, while distributing resources equally.

5. public transport operation planning: public transport authorities consider user equity when optimising bus and train schedules based on passenger needs and the number of passengers. Increasing the number of services at times of high ridership or providing services to specific areas can improve overall convenience.

6. sporting event ticketing: pricing is based on demand forecasts in the sale of tickets for sporting events. The Kelly Criteria can be used to determine how much money to allocate to tickets in order to make the most profit, thereby maximising revenue.

These methods are powerful tools for achieving an efficient and equitable allocation of resources.

reference book

Reference books on the Kelly criterion and fairness-aware optimisation algorithms are discussed.

Reference books on the Kelly criterion:

1. “Fortune’s Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street” – William Poundstone

2. “The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market” – Ed Thorp

3. “The Mathematics of Gambling” – Mason Malmuth

Reference books on equity and optimisation:

1. “Fair Resource Allocation and Rationing at the Bedside”

2. “Algorithmic Game Theory” – Tim Roughgarden

Reference books on statistics and optimisation:

1. “Introduction to Operations Research” – Frederick S. Hillier, Gerald J. Lieberman

2. “Optimization by Vector Space Methods” – David G. Luenberger