Using difference equations to find optimal tax structures on the SugarScape

Abstract

The use of equations to describe agent-based model dynamics allows access to mathematical theory that is not otherwise available. In particular, equation models can be effective at solving optimization problems—that is, problems concerning how an agent-based model can be most effectively steered into a particular state. In order to illustrate this strategy, we describe a modified version of the well-known SugarScape model and implement taxation. The optimization problem is to determine tax structures that minimize deaths but maximize tax income. Tax rates are dependent upon the amount of sugar available in a particular region; the rates change over time. A system of discrete difference equations is built to capture agent-based model dynamics. The equations are shown to capture the dynamics very well both with and without taxation. A multi-objective optimization technique known as Pareto optimization is then used to solve the problem. Rather than focusing on a cost function in which the two objectives are assigned weights, Pareto optimization is a heuristic method that determines a suite of solutions, each of which is optimal depending on the priorities of the researcher. In this case, Pareto optimization allows analysis of the tradeoff between taxes collected and deaths caused by taxation. The strategies contained here serve as a framework for a broad class of models.

Appendices

Appendix 1: Overview, design concepts, and details (ODD) protocol for the SugarScape gradient model

This appendix contains the Overview, Design Concepts, and Details (ODD) protocol for the agent-based model examined in this paper. This protocol was introduced by Grimm et al. (2006, 2010) as a template for describing an agent-based model in detail sufficient enough that the model (and results) can be replicated by an independent researcher. The protocol also includes a list of modeling features that the ABM contains. The description in this appendix is meant to stand alone; as such, overlap may occur between the description here and that contained in the body of the article.

1.1 Purpose

The SugarScape Gradient model is a simplified version of SugarScape (Epstein and Axtell 1996), a model in which abstract entities (ants) roam a landscape made of sugar. Ants all have the same vision strength, but they have varying metabolisms, affecting the amount of sugar burnt over time. Ants that are able to make it to the sugar-rich regions on the right side of the map tend to survive, while those who do not are more likely to run out of sugar and die. Additionally, ants are periodically taxed for their sugar stores—tax rates may be $0, 25, 50, 75\,\%$, or $100\,\%$, and the rates are set regionally (hence regions with more sugar may have higher taxes, for example). Each tax cycle, tax rates may change. The purpose of this model is to investigate the effects of various taxation policies on tax income and ant population. In particular, the model is used to investigate the following question: what is the optimal taxation policy for maximizing collected income while minimizing deaths?

1.2 Entities, state variables, and scales

1.2.1 Grid cells, spatial, and temporal scales

The landscape is a $48 \times 48$ grid of cells containing various levels of sugar. Figure 8 presents a snapshot of the landscape; labels refer to the amount of sugar contained on each grid cell in that region. The landscape wraps vertically but not horizontally; thus, the only way to reach a region with more sugar is by traveling to the right. When ants consume the sugar from a grid cell, the sugar is not depleted. Hence sugar levels are preserved throughout each simulation. Given the fairly abstract nature of the model, time and space are unitless. A simulation consists of a finite number of time steps. At regular intervals, taxes are either collected or not, depending on the tax policy being simulated.

Fig. 8

Landscape of the SugarScape gradient model

Taxation. Taxes are collected every $5$ time steps for a total of $10$ tax cycles, and tax rates may be different in each region and at each tax cycle. Hence, using the values in Table 3, a tax policy is a $4 \times 10$ matrix with entries in $\{0, \, 0.25, \, 0.5, \, 0.75, \, 1\}$. Entry $(i, j)$ of this matrix indicates the tax rate in region $i$ during the $j$th tax cycle. Thus there are a total of $5^{40} \approx 9 \times 10^{27}$ possible tax policies. For each policy, the resultant number of deaths and the amount of tax collected are stored.

Table 3 Taxation and temporal variables

1.2.2 Ants

Each ant has a fixed vision level of $1$ grid cell for the duration of the simulation: this means ants can see exactly one cell in each of the four principal directions up, down, left, and right (unless they are on the horizontal edge of the map, in which case vision does not wrap). Additionally, each ant has a fixed metabolism which determines how much sugar it loses each time step. Movement is governed by vision: an ant moves to the grid cell within its vision with the maximum amount of sugar. Multiple ants may occupy the same grid cell; each ant obtains the full amount of sugar at that location (i.e., sugar is not depleted and there is no exclusion principle in effect). Ants die if their sugar level reaches zero. Ants may accumulate up to $50$ sugar (Table 4).

Table 4 Ant state variables

1.3 Process overview and scheduling

Per the suggestion in Grimm et al. (2010), the ABM process is presented in Algorithm 2 as pseudo-code in order to give a detailed account of the order of events. The ant and tax routines are executed fully by one ant, then fully by another—i.e., serially. This means that state variables are updated asynchronously. Time steps are discrete units, as is movement: ants jump directly from the center of one grid cell to the center of another.

1.4 Design concepts

1.4.1 Basic principles

The SugarScape Gradient model builds on SugarScape, which can be studied in a myriad of contexts, including sociology, ecology, economics, and culture. The primary motivation for this version is to investigate the effects of various tax policies in an abstract setting. It is hoped that the model will provide insight into the relationship between tax policies and population dynamics, and that these insights might be extrapolated to some real-world system.

1.4.2 Emergence

Spatial population dynamics ought to be an emergent property of SugarTax: for example, frequent high taxation in high-sugar regions might substantially alter regional population counts. The precise mechanism driving such changes is not built in to the model in any direct sense.

1.4.3 Objectives

The objective of each ant is to move to a cell within its vision with the maximum amount of sugar. There is no other consideration, and ants do not have knowledge of past or future taxation in any location.

1.4.4 Sensing

Ants are aware of the sugar level and occupancy of each grid cell within their vision. They are not aware of any properties of any other ants, including those within their vision.

1.4.5 Stochasticity

Ant movement is partially stochastic: if there are multiple grid cells within an ants vision with the same (maximum) sugar level, the ant will move to one of these grid cells at random. This stochasticity is vital to population distributions—if the ants chose the nearest such cell, for example, populations tend to cluster on the boundary of each region. Clustering severely affects migrational patterns, as ants are not encouraged to explore the landscape. This movement feature is discussed in Epstein and Axtell (1996).

1.4.6 Observation

Each simulation occurs over a finite number of time steps. At each time step, the following information is collected: the number of ants in each region, the wealth distribution of the ants in each region (separated into disjoint bins), the total amount of tax collected, and the number of deaths occurring in each region. In addition, the tax policy implemented over the course of the simulation is recorded as well. At the end of each simulation, these data are written to a comma separated value (.csv) file, a universal format for spreadsheet applications.

1.5 Initialization

The model is initialized with an average of $62.5$ ants; each is placed at a random location on the landscape. Ants begin with a random amount of sugar between $5$ and $25$ (inclusive); this value is different for each ant and chosen from a uniform distribution. Ants are initialized with vision $1$ and metabolism either $1$ or $2$. Vision and metabolism of a given ant do not change over the course of a simulation.

1.6 Input data

The landscape is read in from a.txt file; this helps with implementation and makes it easier to make changes. A tax policy can either be chosen directly via code manipulation or chosen at random. The policy must be chosen prior to simulation. As such, the tax policy may be thought of as input to the model.

Appendix 2: Difference equation models

Figure 9 contains the population equations for the $48$ columns in the ABM; Fig. 10 contains the wealth (i.e., sugar level) equations. Note that in order to match ABM behavior, all equations are updated synchronously. See Table 1 for the order in which equations are updated. Note that since there are multiple updates of the equations in order to simulate one time step in the ABM, the time scale in the equation system is not quite the same. In particular, time steps in the equations are more closely aligned with processes in the ABM (for example, agent movement or taxation).

Fig. 9

Population equations

Fig. 10

Wealth equations. Note that these equations track sugar levels based on sugar eaten and sugar metabolized, but do not account for migration or taxation. Here, the number of ants with $0$ sugar is always set to $0$; this is done to simulate death. Note that according to Table 1, wealth values are updated after migration and population values are updated after taxation. This ensures that at the end of each time step, the sum of the population equations is always the same as the sum of the wealth equations. Tax rates are percentages, but a ceiling function is applied in order to ensure that an integer amount of tax is paid

1.1 Initial values

Initial values for each of the equations are provided in Fig. 11. These values are determined by settings from the ABM. Note that in this study we focus only on agents with low vision (i.e., vision $1$) and low metabolism (metabolism $1$ or $2$). In the original ABM, there were three other categories of ants, total of $250$ inital agents. Hence, since we are focusing on one category, the initial population values must sum to $250/4 = 62.5$. These are placed at random throughout the grid. Likewise, ants begin the simulation with an amount of sugar chosen uniformly randomly between $5$ and $25$ (inclusive); hence the number of ants in a region determines the expected number at each sugar level.

Fig. 11

Initial values. $48$ population equations track the number of ants in each column. There are four regions and $21$ possible starting sugar amounts ($5$–$25$ inclusive); the $62.5$ initial agents are assumed to be distributed among these regional sugar levels uniformly. Wealth equations hold for $r \in \{1, \ldots , 4\}$ and $m \in \{1, \, 2\}$