Agent-Based Modelling of Future Dairy and Plant-Based Milk Consumption for UK Climate Targets

: A reduction in the production and consumption of meat and dairy across much of the world is critical for climate change mitigation, the alleviation of ecological stress, and improved health. We update an agent-based model (ABM) of historic UK milk consumption and apply it to scenarios of dairy reduction and adoption of plant-based milk (PBM) out to 2050. The updated model is comprised of a cognitive function, where agents perceive the physical, health and environmental characteristics of milk choice, which is modified by habit and social influence. We use European Social Survey 2018 and British Social Attitudes 2008 survey data to empir-ically inform the model. Taking a backcasting approach, we calibrate parameters against published UK dairy reduction targets (2030 and 2050), and test how different price relationships, and characterisations of environmental concern, may affect simulated milk consumption from 2020 to 2050. Scenarios for core targets (20% less dairy by 2030 and 35% by 2050) largely produced plausible consumption trajectories. However, at current pricing of dairy and PBM, simulated consumption was mostly unable to deliver on desired core targets, but this improved markedly with dairy prices set to organic levels. The influence of changing environmental concern on milk choice resulted in higher levels of dairy milk reduction. When modelled as transient, intense shocks to public concern, consumption patterns did not fundamentally change. However, small, incremental but permanent changes to concern did produce structural changes to consumption patterns, with dairy falling below plant-based alternatives at around 2030. This study is the first to apply an ABM in the context of scenarios for dairy reduction and PBM adoption in service to UK climate-related consumption targets. It can serve as valu-able bottom-up, alternative, evidence on the feasibility of dietary shift targets, and poses policy implications for how to address impediments to behavioural change. different representative price relationships between dairy and PBM; and modelled different mechanisms for changes to agent environmental concern and milk choice influence.


Introduction
. The global food system is at the intersection of several connected crises, imposing a severe planetary burden that necessitates transformational change across all aspects of production and consumption. One key component of this is the need for dietary reduction of meat and dairy across much of the world. The latter is the focus of this study, and is responsible for around . Gt CO 2 eq (GDP ), as well as stress on water, land, and ecosystem pollution (Mekonnen & Hoekstra ; Poore & Nemecek ). However, there is heterogeneity in the overall scale of impact of dairy milk at regional, national, and farm level, depending on factors such as geography, farming and production method (Guerci et al. a,b; Poore & Nemecek ). Further, globally some focusing on individual decision-making and behaviour, rather than aggregate groups. ABMs can capture heterogeneity and incorporate diverse empirical data, for example in attitudes, preferences, biases, habits and demographics across populations. It is distinguished from similar techniques, such as microsimulation, by the possibility of social interaction between agents. A key feature of ABMs is the possibility for emergent macrolevel outcomes that cannot be predicted or constructed simply by the individual parts of the simulation. This makes them well-suited to explore macro-level paths for future (more) sustainable diets, by investigating individual food choice and consumption behaviour. .
ABMs have been employed to investigate healthy diets (Auchincloss et al. ; Zhang et al. , ), sustainable food (Lloyd & Chalabi ; Namany et al. ), and have gained traction as a tool in wider public health research (see Tracy et al. for review). For the most part, these investigations have focused either on individual dietary behaviour, or on the sustainability determinants of food systems at the supply level. Fewer have combined these avenues to explore sustainable diets, although this is now receiving greater interest, with studies that for example have assessed meat consumption behaviour of UK consumers (Scalco et al. ), and looked at the impact of global trade and climate change of food and nutrition (Ge et al. ). .
We use a previously developed ABM of food choice that has been used to investigate and reproduce historic UK milk consumption trends (Gibson et al. ). Here, we update the model and apply it to a forward-looking analysis of UK dairy and plant-based milk consumption. This ABM uses the conception of food choice influences from Chen & Antonelli ( ), incorporating aspects from each of food-related features, individual di erences, and society-related features. Specifically, price (Andreyeva et al. ; Annunziata & Scarpato ; Baudry et al. ; Hoek et al. ), health (Kang et al. ; Verain et al. ), environmental/sustainability concerns (Ricci et al. ; Verain et al. , , ), habit (van't Riet et al. ), and social influence (Cruwys et al. ; Higgs & Thomas ; Pachucki et al. ). Drawing on and operationalising empirical survey data, agents di er in the basic values they hold and the relative importance they ascribe to these di erent food choice influences.

Model Description
. The core agent-based model that we update and apply in this study was developed in Gibson et al. ( ) using the ODD (Overview, Design concepts, Details) protocol, a de-facto standard for documenting agent-based models (Grimm et al. , , ). Here, we focus on the aspects of model development and other specific elements of relevance, but provide additional details in Appendix B. The framework for agent decision-making is given in Figure . Overview Purpose .
The model's overall objective is to produce consumption trajectories from out to for dairy and plantbased milks (PBM). It does this by modelling individual-level preferences and food influences (informed by both theoretical grounding and empirical data) across physical, health and environmental perceptions, habit, social influence and the active evaluation of prior-choice. These future consumption curves (reported in average ml of milk per person per week) are directed, through parameter calibration via optimisation, to try and meet dairy reduction targets posed by UK bodies for and . Specifically, the study performs simulation experiments to assess and compare six di erent milk consumption scenarios that are distinguished by di ering model assumptions and target level.
Agents represent consumers that each have a disposition to consider (or not) their milk consumption choices.
Agents construct a cognitive choice function for dairy and PBM, comprised of the perceived physical (modelled as price), health and environmental characterises of each choice. These are computed at each time step of the simulation, and are modified by other food influences (habit, social influence) and choice evaluation, each governed by individual sub-models. The relative importance that agents ascribe to the physical, health, environmental, habit and social influences is determined by empirical data operationalised from the British Social Attitudes (BSA) survey (National Centre for Social Research ). The use of survey data to construct . The key functions that act upon agent choice are described in the sub-model section. .
At each time-step, the milk consumption of agents that are 'disposed' to consider their choices is given by the relative proportion of each option's choice function of the total summed choice function, multiplied by the total milk consumption. For instance, if dairy and PBM have the same choice function value, an agent will consume % of the total available milk (expressed as ml per person per week) for each choice. Agents that are not in a state of disposition at a given time-step repeat their previous choice and milk consumption.

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In the model, total average weekly consumption was maintained at levels over the simulation period out to . This was because we were primarily concerned with product substitution rather than absolute decrease in consumption. However, this is a clear motivation for future work, and as a starting point, Figure in Appendix A shows a simple extension to the scenario analysis by considering future non-constant (declining) total consumption.

Implementation and initialization
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Input data
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Agents are initialised with basic human values (universalism and security values; Schwartz , , ) using data operationalised from UK specific responses (n = 2, 167) of the European Social Survey (ESS) (Norwegian Centre for Research Data ). The survey questions associated with these values are reproduced in Table of Appendix A. Survey responses were on a six-point scale of 'Very much like me' to 'Not like me at all' and also included an additional three coding options of 'Refusal', 'Don't know', and 'No answer'. Specifically, we take cross-tabulated data of the two relevant question responses, weighted to account for di erences in selection probability from the sampling design, and obtain the proportions that cover each of the possible response combinations. A random sample of these responses is taken, equal to the number of agents modelled. Here, this was typically , , which was tested with resampling and di erent agent population sizesee Appendix A. The sample is loaded into the model and each agent is assigned a universalism and security attribute accordingly. A uniform probability distribution determines the specific number that this takes, with the six-point response scale converted into six equal sized bins between and . E.g., an agent with a 'Very much like me' response will have an equal probability of scoring between . and . , and so on. .
Agents are also assigned a weighting for each of the main food influence categories in the model (physical, health, habit, social, environmental). These weightings are operationalised from British Social Attitude (BSA) survey data which included a series of questions on food influences, and assigned directly to agents at individual level. BSA contains , survey responses across a number of social attitude and demographic dimensions. This study was interested in the section on food influence, which contained direct influences (and options for 'other', 'someone else decides' and 'no particular' influence). Responses were recorded as either (having an influence), (not having an influence), or -(did not answer). A er removing null responses (values), , responses remained, of which , were randomly sampled for inclusion in the model to directly represent agent influences (as with ESS, multiple samples were tested at , and di erent samples sizes were drawn -see Appendix A). The di erent influences were assigned to one of five most closely aligned categories (physical, habit, health, social, environmental), and then each category was scaled, so that categories had the same total representation, summed and converted into a proportion of the total summed influence. The mean weights across the sample were: physical = .
, habit = . , health = . , social = . , environmental = . . See Table in Appendix A for details of the BSA survey questions.

Sub-models
Disposition . Gibson et al. ( ) compare two mechanisms of disposition, a threshold-based, and a probability-based approach. From that study, it was found that the latter performed better in reproducing observed macro level data of historic milk consumption. And so, here, we opt to employ the same probability-based disposition approach, which itself was influenced by previous studies modelling agent disposition dynamics in social networks (Galán et al. ; Wang et al. ). The probability to become disposed to consider milk choice options is based on how alike an agent's neighbour choices are, and uses information entropy to calculate maximum and minimum 'alikeness'. Equation expresses this disposition function: Table ) is the gradient of the probability logistic function; . is a coe icient to limit values between and ; and h hmax gives a proportion of how homogenous or heterogenous an agent's neighbours aggregate choice is (see Equation ), where h max equals (−log 2 0.5). Equation expresses neighbourhood choice information entropy: where, f dairy and f P BM , are the frequency of an agent's neighbours that choose dairy milk or PBM, and f all is the total number of neighbours.
The cognitive perception sub-model represents how information regarding di erent milk choice characteristics are perceived by agents. Central to this are the calibrated health and environmental perception parameters, the value of which is varied to reflect its non-constant nature. I.e. perception of something can change with time, space, context, and of course di erent individuals. Values are drawn from a normal distribution, where the means of these distributions are determined by the perception parameters, with standard deviation of . . A normal, rather than say a uniform, distribution is chosen as it gives a higher and symmetric probability of producing a value close to the calibrated mean, while still allowing the chance of values to deviate strongly from this.

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For scenarios that consider current and organic pricing, means are taken directly from a fixed value representing the relative price relationship between dairy and PBM. Price data on PBM and organic dairy was collected online (in November via manual means) from publicly available data from three major UK supermarkets (Tesco, Sainsbury's, Morrisons). Average prices for conventional dairy milk were calculated from UK Family Food Survey / data. Mean values were; p/l for PBM, p/l for organic dairy, and p/l for conventional dairy. Note, prices for PBM and organic milk were simple averages and not weighted by product volume sold.

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These prices were operationalised so as to enable adequate inclusion in the cognitive function. Here, a larger price is a negative characteristic, and the model treats overall choice as a positive sum of all the di erent influences. The PBM price was set at (most expensive, therefore lowest score) and the price multiplier between PBM and current conventional milk or organic milk was assigned to these options accordingly. This resulted in a value of . for current conventional milk, and . for organic milk. That is to say, PBM is . times more expensive than conventional milk, but this is represented as a positive 'bonus' for dairy. This may not be the optimal approach if we were concerned with more granular realism, but for the purposes of this study, this abstraction was deemed a reasonable proxy. .

The cognitive choice function (Equation )
is comprised of the three modelled milk characteristics, weighted by the relative importance placed on it (out of the five influence categories assessed).
where β phy , β hel , and β env are the weights assigned to the perception of physical, health, and environmental aspects of the milk choices. At initialisation, agents are assigned a set of weights drawn from BSA survey data (see 'Input data' for more details).

Environmental concern .
In this function, exogenous changes to agent environmental-based choice influence are modelled. It consists of two variables: a probability of occurrence, and a magnitude of e ect. The two di erent approaches that scenarios S and S test are constructed from YouGov weekly/monthly public concern issue tracker data (YouGov ). Here, we approximate the longitudinal change in UK public environmental concern (given as a % of people that rank 'environment' as a top issue) as the size of potential percentage change in environmental weighting of milk choice influence (β env from Equation ). This percentage change in weighting is added to an agent's existing environmental weight, and subtracted from its physical (price) weight (β phy ). This ensures that the total influence weighting remains equal to , with the model controlling for any weight values that would be outside of the to range. .
In the case of scenario S , the probability of a shock occurring was based on the instances of clear and discrete concern spikes that have occurred over the data range ( -). Over this -year period, three such instances occurred, that coincided with the severe UK flooding of , Extinction Rebellion protests in , and the start of COP in November . From this, concern shocks were approximated as having a / or % chance of occurring on any given time-step in the model. The size of this e ect was given by the average percentage change in concern between the start of a year and the point at which a spike occurred, which was calculated at %. If a concern shock occurs, the new agent influence weights feed through the model and agents make choices based on these updated values. At the end of the decision-making process and time-step (year), this e ect is reversed to mimic the temporary nature of such concern shocks. .
Scenario S follows a similar procedure, however, the probability is set a , to reflect the continuous nature of increasing concern. The size of this e ect was modelled as the total annualized observed change in concern from to December (latest tracker data). To account for unequal distribution of tracker data, an e ective daily value was calculated that was then annualized to give . %.

Habit .
In the model from Gibson et al. ( ), habit was treated as a multiplier to subsequent choice function scores that had repeatedly returned the highest value of the options available. That is, if a choice function of a given milk option consistently scored higher than the other option, eventually the habit bonus would trigger, further entrenching this option. We take the same form of this habit function, i.e., the empirical function of habit formulation from Lally et al. ( ), however, in this study it was applied slightly di erently. Here, it was additive rather than a multiplier, to ensure internal consistency with how the other four influence categories are modelled (physical, health, environmental and social). That is, a mixed additive and multiplicative weight and influence construct could yield disproportionate weight e ects to their values. For instance, if one weight is added but another multiplied, this could increase or decrease their relative contribution, deviating from their assigned proportions. And so, to avoid this we followed a wholly additive approach. Further, its total impact is modulated by the weighting a given agent ascribes to the influence of habit on food choice. This is detailed by the following equation: where peak habit (fixed at two, but future model iterations should examine this with robustness analysis) is the maximum influence that habit can exert, consecutive choices is the number of repeat highest scored milk .
This study adapts the original formalism of social influence employed by Gibson et al. ( ). As with the original model, an agent has a probability of interacting, where influence is modelled as the mean set of choice functions across its neighbour network. However, instead of a free parameter that was termed 'social susceptibility', the extent of this neighbour influence is governed by the weighting an agent ascribes to social food influence (operationalised from BSA survey data). This is represented by the following equation: where f (cog.) mean neighbour is the average value of neighbour cognitive choice functions and β soc is the weight assigned to social influence.

Total choice function .
The total choice function for each option is then given by the weighted sum of each influence component, expressed by the following equation: Agents have the opportunity to evaluate, learn from, and inform their future milk choices. This function remains largely intact from Gibson et al. ( ), employing a conceptualisation of cognitive dissonance between an agent's human values (from ESS survey data) and the impact of their milk choice behaviour (see Table ). The minor update in this study is that agents now also look to minimise or escape a state of cognitive dissonance by altering the weight (± % per time-step) they ascribe to health and environmental components versus physical (price) aspects. This is an e ort to further draw on the empirical data from BSA .

Simulation Experiments, Calibration and Analysis
Scenarios and simulation experiments . The overall objective was to assess, through two sets of simulation experiments, the feasibility of possible UK milk consumption trajectories out to under di erent scenarios and target levels. A backcasting-type approach was followed, whereby scenario parameters were calibrated (see next section) to try and produce 'desired' dairy reduction targets posed by the UK's CCC and Eating Better alliance. The first set of experiments explored rising dairy milk prices, the second looked at how increasing environmental concern may reduce dairy milk and increase PBM. Specifically, six scenarios (see Table ) were constructed that looked at: a) di erent price combinations (current and organic dairy pricing); and b) di erent mechanisms of changing the influence of environmental concern in food choice. These scenarios were considered under di erent levels of target ambition (the CCC's core targets of % less dairy by and % by ; and extension targets of % by , along with the Eating Better alliance's more ambitious % by ). .
The six di erent scenarios are as follows: S , which most closely resembles a baseline, looks at a current price relationship between dairy and PBM, with targets of % less dairy by and % by ; S instead considers rising dairy prices set at average organic levels, again with targets of % less dairy by and % by ; S and S repeat these price relationships but look at the more ambitious targets of % by and % by . These four scenarios do not explicitly consider significant changes to the environmental concern basis of agent food choice, beyond the modelling of influence weights and individual values endogenous to the ABM. That is, the ABM represents the relative influence of di erent food choice aspects as heterogenous weightings drawn from survey data. These weightings can then dynamically change based on an agent's individual values and the choices they make. .
However, in scenarios S and S , this agent-focused individual-level function remains, but an exogenous 'societalwide' factor to change environmental concern is introduced and tested. In scenario S , this is represented by a stochastic intense 'shock' that temporally shi s agent's choice influence weighting toward environmental concern. Scenario S has a di erent representation, looking at a smaller, incremental but sustained, increase toward greater environmentally weighted influence. More details are given in the Model Description and 'Environmental concern' sub-model sections. All scenarios and model runs had the same initial conditions of , agents, average dairy milk of . ml and PBM of . ml per person per week, which ran at yearly intervals from to . Parameter values and ranges are those given in Table of  CCC core: % less dairy by and % less dairy by S Current Exogenous, sustained and incremental increase to concern CCC core: % less dairy by and % less dairy by Table : The scenarios constructed and implemented to model UK dairy milk and PBM consumption paths out to .
Calibration and uncertainty analysis .
Calibration was performed for each scenario over the bounded range of parameters (see Table ). The optimisation exercise tried to minimise the absolute di erence between the simulated consumption levels from scenario outputs with that inferred by the target level at or . That is, a set of model parameters were calibrated at the micro level to give macro level consumption outputs that tried to meet UK dairy reduction targets. The general computational technique follows that employed in Gibson et al. ( ), and is summarised briefly here. An optimisation exercise based on an evolutionary algorithm (EA) (Python's DEAP package) first generated a population of parameter sets (n = 75), sampled using a uniform distribution over each parameter's bounded range. The EA then started an 'evolutionary loop' on this population, producing children (n = 75), evaluating the fitness of both population and children, and then selecting (n = 75) the most fit candidates to become the population of the next generation. The loop ran for generations. The fitness criteria were the minimisation of the absolute di erence between the simulation output and target consumption level at and . Given the bi-objective nature of the optimisation, the EA produced a pareto-front of n candidate parameter sets for further evaluation. Owing to time and resource constraints, we did not conduct hyperparameter optimisation or 'tuning' of the EA process (i.e. through grid search or even another genetic algorithm). Initial conditions were set at , agents, with a starting average dairy milk consumption of . ml and PBM consumption of . ml per person per week, which ran at yearly intervals from to .
. From these results, uncertainty analysis was conducted to simulate model outputs (milk consumption curves). Saltelli sampling was used to generate parameter values over a ± % range of the calibrated parameter sets for each scenario. Sample size was given by the expression n(2p + 2), where n is the baseline sample and p the number of model parameters. Following the general sample size adopted in Gibson et al. ( ) we sample and run each parameter set times. To assess the appropriateness of this value, i.e. is it under, over, or adequately specified, we calculate the required sample size using a confidence interval approach set out by Byrne ( ). Here, sample size is given by the expression (z α 2 * CV /w) 2 where z α 2 , the standard normal, is . for % confidence, CV is the coe icient of variation, and w is the target confidence interval width expressed as a proportion of the mean, and set at . . This expression is applied at each time-step from to for both dairy and plant-based milk simulation output. Calculated sample sizes were as follows: dairy (mean = 62, max = 101), plant-based (mean = 307, max = 470). From this, the initial sample size of is deemed a reasonable number, being comparable to the mean across plant-based simulation results, and significantly larger than the requirements for dairy. Each of these sample parameter sets were run by the model from to , and the outputs and performance compared against target levels.

Sensitivity analysis .
Temporal global variance-based sensitivity analysis (TGVSA) was performed on the most representative (closet to the mean) PBM model run for each scenarios S -S . Scenarios S and S used the calibrated parameters from S and were not considered in the sensitivity analysis. Sample size (n = 1, 064) followed the minimum threshold set out in Gan et al. ( ), drawn over a ± % range of the central value. Continuous TGVSA was computationally prohibitive, and so six time-step instances ( , , , , , ) were selected. Python's SALib package (Herman & Usher ) was used to conduct Sobol analysis with Saltelli sampling to assess the relative contribution of each parameter to total sample variance. Variance based sensitivity methods such as this can in principle fully decompose output variance due to model parameters and all combinations of interaction. However, quantifying specific higher order sensitivity indices (e.g., second and third) is computationally expensive and researchers typically limit analysis to a quantification of first-order (S) and total-order (ST) (Saltelli et al. ). ST can be thought of giving an aggregate of all variance, and if S is known, then a synthetic value of all higher-order e ects (interactions) can be calculated. Further, this approach is capable of dealing with non-linearity, which is common to these types of models.
. Other experimental designs for global sensitivity analysis of multiple parameters are available (e.g., linear models, fractional factorial, Latin Hypercube Sampling (LHS)). Linear models and fractional factorial methods are not generally appropriate for models with nonlinear e ects (Saltelli et al. ). LHS would have been a possible approach, but we opt for the extended Sobol' due to its coverage of the parameter space (for S and ST) in as fewest samples as possible. In this sense it has been regarded as superior to LHS (Kucherenko et al.

Calibration
. Thirteen parameters were calibrated against objective functions that looked to minimise the absolute di erence between reduction target and simulated milk consumption level for the given target year. That is, each parameter was not individually calibrated but rather, calibration was performed on the model output resulting from parameter values. An optimization approach via evolutionary algorithm was employed to conduct the calibration exercise. Table shows a Table : Parameter calibration results for each scenario. Multiple optimised parameter sets under S ,S ,S ,S are summarised by mean values and standard deviations (in parenthesis). S -S produced one set of parameters from a single objective calibration. 'a' shows results from calibration against a target level, 'b' shows the same for a target level.
. Scenarios S and S used the calibration results from S (Table ). Mean values and standard deviation (in parenthesis) of calibrated parameter sets are shown for the bi-objective (multi-target) optimisation scenarios (S ,S ,S ,S ). Under S -S , 'a' is the result of the target and 'b' refers to the objective target level. .
Comparing parameters , and , for scenarios S -S (highlighted and in the table), shows that calibrating for a % consumption reduction level generally produced a highly favourable perception of PBM relative to dairy for health and environmental impact. Care must be taken when interpreting these calibration results. Summarising multiple di erent parameter sets has its limitations in that mean values within, and across, scenarios are not directly comparable due to heterogeneity across sets and some parameter dependence. Further, as with many agent-based models, the interactions between parameters are o en more influential than the parameter values themselves.

Simulation Experiments
Increasing dairy milk price . Figure shows simulated dairy milk (navy lines) and PBM (pink lines) consumption from to for each parameter set of each scenario. Historic consumption is plotted from to . Target levels are shown for a % reduction by (solid black line), % reduction by (dashed black line), and % reduction level (dash-dot black line). For scenarios S and S (bottom row), both the (navy and pink) and (dark and light grey) target results are shown. The le -hand side shows results of milk pricing represented at current levels, the right-hand side considers if all dairy were priced organically and PBM remined constant. Figure : Results of model simulation of UK dairy (navy) and PBM (pink) consumption for each calibrated parameter set of each scenario. Target levels are shown for a % reduction by (solid black line), % reduction by (dashed black line), and % reduction level (dash-dot black line). Scenario S show results for current relative pricing between dairy and PBM, S consider if all dairy milk was priced organically, and S -S repeat this di erent pricing but for the % reduction targets by or . .
Overall, the calibration-optimisation exercises succeeded in generating parameter sets that were able to produce model runs in the region of the desired target level. Figure explores this in more detail. Scenarios S -S show consumption patterns that o er reasonable trajectories for dairy milk reduction (with concomitant increases in PBM). This contrasts with the results of S -S which display a rapid decrease in dairy (and rise in PBM), and then a plateau from around onwards. The rate of change produced by these results to achieve either the % by or target would appear implausible, though not impossible.
. To the le of the red line are scenario runs that 'surpassed' the dairy reduction target (i.e., resulted in a consumption level below that of the target for the given year). To the right are scenario runs that 'failed' to meet the target (i.e., resulted in a consumption level above that of the target for a given year). The percentage and absolute number of scenario simulation runs that met the target are shown for each heatmap. Note, scenarios have di erent total numbers of runs as the target optimisation calibration analysis produced multiple candidate parameter sets. Figure a shows this for one 'half' of the bi-objective target exercise -% reduction in dairy by . Figure b shows the other 'half' of this -% reduction by . Figure c shows the results of a more ambitious % reduction by , and Figure d shows this same target level for . For Figures c and d, single target optimisation calibration was conducted, producing a single parameter set. As a result, they have fewer total runs. Figure : One-dimensional heatmap 'trace' of the distribution of scenario simulation runs that surpassed of failed the given target (red line). a shows this for the % less dairy by target, b shows % less by , c shows % less by and d shows % less by .
. Scenario S , where relative pricing was fixed at organic dairy to PBM levels ( p/l for organic vs p/l for conventional and p/l for PBM), produced significantly higher proportion of successful runs than S (current conventional pricing). Here, the rise in average dairy pricing lead to an increase in 'successful' runs of . % to . % for and . % to . % for . Scenarios (S -S ) that looked at more ambitious targets ( % by or ) mostly failed to meet or surpass the reduction level. Scenario S (current pricing) in particular did not produce a set of calibrated parameters that would give the required target level. However, S was close to this threshold, an expected result given this was the output of a single-objective optimisation exercise.

Shi ing environmental concern .
Figure shows the output of scenarios that explored two di erent mechanisms of change to the environmental weighting of milk choice. Scenario S (same as in Figure ) is given for comparison. Scenario S and S take the same calibrated parameter sets as S and simulate consumption curves to test the impact of each of these mechanisms. The first, S , modelled temporary random environmental concern shocks, while S modelled a sustained, incremental and permanent shi toward greater environmental based milk choice influence.
. Compared with S , simulation outputs for S and S resulted in a larger decrease in dairy milk consumption (and higher uptake of plant-based milk). However, when comparing S and S , the latter shows a markedly di erent output, with PBM surpassing dairy milk consumption at around and % of model runs hitting and targets from the CCC. Figure : Results of simulation experiments to model scenarios of shi ing environmental concern in milk choice. Dairy is given by navy lines, PBM by pink. Target levels are shown for a % reduction by (solid black line) and % reduction by (dashed black line). S shows results with the introduction of temporary environmental concern shocks. S shows the same but for sustained, incremental increases to concern. Both scenarios use calibrated parameters from S , which is shown for comparison.

Discussion
. Overall, scenarios produced plausible consumption trajectories for the core Climate Change Committee target dairy reduction levels of % by and % by . Both sets of experiments, rising dairy milk prices (to organic levels) relative to plant-based alternatives, and changes to environmental concern, resulted in larger decreases in dairy consumption than the baseline scenario S . Within this, the largest impact on simulated consumption trajectories was found with the introduction of small, but permanent, incremental increases to agent environmental concern, and the impact this had on influencing milk choice. Meeting extension targets of % dairy reduction by or (S and S ) appeared more di icult to achieve, with simulation trajectories that, on the face of it, would seem challenging to reproduce in the real-world. .
Scenario S , which most closely resembles a current baseline, did not produce a majority of runs that met and targets (just . % and . % respectively). As a reminder, scenario S contained the core ABM decision-making and influence functions that govern model milk choice. These consist of: cognitive perception of impact, social influence, habit, and choice evaluation, i.e. internal individual-agent factors that determine consumption options and the observed calibrated substitution curves. S did not include the two external factors tested in the experiments. In other words, absent of the explicit price changes and environmental-concern shi ing interventions tested by this study, model simulations did not overwhelmingly deliver the CCC's base case dairy reduction targets. However, all things being equal, representing all dairy milk at average organic prices (S ) showed a marked increase in successful model runs hitting, or surpassing, target levels. In this scenario, . % of runs met a % reduction by target, and . % met the target ( % reduction). Clearly, the wider land and sustainability impacts of shi ing UK milk to % organic has not been explored here, however, we use 'organic pricing' as an indicator of the more reflective actual costs of dairy milk (in the absence of robust, but emerging, true cost accounting). An implication for policy should be to consider how pricing structures can be used to motivate reduction of dairy consumption and uptake of PBM. .
Scenarios (S and S ) were largely unable to produce a set of simulations that met the more ambitious targets of % by either or . This was particularly the case of S , which could have implications for the feasibility of achieving deep shi s to sustainable diets in the absence of clear interventions. While scenario S also largely failed to meet the stated targets, it was, however, far closer, and these results demand more considered interpretation. Unlike scenarios S and S that conducted bi-objective optimisation, this set of scenarios produced a single set of parameters and therefore had less heterogeneity and uncertainty to explore. It is entirely possible that calibrating via single-objective optimisation could produce a lower (even more ambitious) target. If the metric for performance is relaxed, we see that almost all of these runs are less than % short of their target level. .
Scenarios S and S looked at how the introduction of random temporary shocks, or, smaller but sustained increases to agent environmental concern, may impact milk choice and consumption. Shocks to environmental concern with the UK public have occurred in response to severe discrete weather events, as was the case for the floods, or more recently, due to increased societal visibility and attention to climate change (Extinction Rebellion (XR) and COP ). However, environmental concern receded rapidly post-flood in , and concern just a er COP had a % negative swing a er reaching historic highs of %. The spike caused by XR protests did not fully revert to pre-protest levels until the onset of COVID-in early , where health issues overwhelmingly topped the data. In general, UK public environmental concern remained stable at around % from to . From October (perhaps a response to the IPCC's SR being released) to date, concern has trended upwards, and is now at around %.

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In modelling these di erent observed public concern phenomena, via agent milk choice influence, we show that both produced larger declines in dairy milk consumption than the S base scenario. However, even with a significant and rapid rise in concern, the transient nature of shocks did not fundamentally alter the emergent consumption pattern. In contrast, small, incremental but permanent increases to concern produced markedly di erent consumption profiles, with dairy and PBM curves actually crossing over in before plateauing from . While recognising that transformative change toward sustainable food systems requires fundamental, structural, measures that go beyond individual behaviour, there are nevertheless implications here for a shi toward more sustainable diets. One way to engender sustained and increasing change is a self-sustaining environmental information ecosystem where policy, responsible media, communities, and everyday lived experience act to mutually reinforce meaningful concern among the public. .
Continuous improvement of both ABM modelling e orts and studies is key to the wider development of the ABM community, and its contribution to science, policy and practice. Giabbanelli et al. ( ) conduct a review of ABMs on obesity, and o er a checklist with which to appraise their quality. The checklist covers five dimensions (data, parameters, sensitivity, validation, documentation and reproducibility) that cover a total of items. The authors construct three tiers (T -best, T , T -worst) based on how a given study performs against each of these items.
. Overall, the model scored mostly T s, but had two T s (recent data, free parameters) and two T s (model design sensitivity analysis, comparing outputs of validation) (see Table in Appendix A for results). The four items that score less than T are briefly discussed. Recent data -BSA survey data is from , which would qualify as T , however, ESS, price and impact data are all from the last years which would qualify as T , and so, we score this element as T . Free parameters -all parameters are detailed, but the study has not attempted to minimized the number of free parameters. Model design sensitivity analysis -sensitivity to model design is an important aspect of model investigation and analysis. Although we do not conduct such an analysis here, we do acknowledge that the model would benefit from this kind of robustness analysis and signpost this as an important avenue of future work. Comparing outputs of validation -it is challenging to validate against data given a forward-looking scenario-based study, however, the base model on which the ABM is developed was validated against historic observed data. Some of these limitations are discussed in more detail in the following section.

Limitations and future research .
Here, some study limitations, as well as opportunities for future research, are briefly described. A central interest of the study was to analyse possible influences and dynamics for the reduction in dairy milk by substitution for plant-based alternatives. Although overall milk consumption has declined for several years, in the model, total average weekly consumption was maintained exogenously at levels over the simulation period out to . Future research could look to incorporate scenarios that looked at substitution along with declining consumption, and assess their interaction and relative impact on overall reduction trajectories (we give an initial exploration of this in Figure in Appendix A). .
The underlying data to construct PBM health and environmental impact is based on a weighted average of the market share of the three most widely consumed plant-based milks in ; soya ( %), almond ( %) and oat ( %) (Kantar ). The model does not account for a changing market share (e.g., increase in oat milk) or the emergence of new entrants (e.g. potato milk). Future research could look to disaggregate these di erent plantbased alternatives and include them as distinct options (with their own health, environmental and physical characteristics) in the ABM. .
Data used to give some empirical grounding to agent influence weightings was from . To the authors knowledge, this is still the latest and most comprehensive set of survey data for UK public food influences, however, at over a decade old, it is likely that these influences have at least in part shi ed. This is somewhat mitigated by a model structure that allows agents to change their influence weights in response to cognitive dissonance or external triggers. However, given the increasing focus on food systems, diets and sustainability, it would be beneficial for an updated survey to be included in the next iteration of British Social Attitudes, or similar large-scale survey. .
Finally, although building on an existing ABM calibrated to historic consumption of UK whole and semi/skimmed milk, this model is still by necessity only one representation of food consumption influence. Future research and modelling e orts should assess, with rigorous robustness analysis, model choices and structure.

Conclusion
. Dietary shi toward lower meat and dairy consumption is a critical lever to mitigate climate change and address wider socio-ecological challenges. Recognising this, various UK bodies (e.g., the Climate Change Committee) have posed dairy reduction targets for and . This study updated and applied an existing ABM of UK milk choice to analyse scenarios for dairy consumption reduction targets, with concomitant increases in plantbased milks. .
Specifically: it enhanced empirical grounding with the introduction and operationalisation of food influence food survey data from British Social Attitudes ; incorporated a set of plant-based impact data; developed di erent representative price relationships between dairy and PBM; and modelled di erent mechanisms for changes to agent environmental concern and milk choice influence.
. Taking a backcasting approach, it aimed to generate and compare several di erent sets of parameter values and model runs, calibrated, via optimisation, to try and produce target levels of reduced dairy consumption. Two sets of simulation experiments were conducted to assess, through the lens of agent-based modelling, the feasibility of possible milk consumption trajectories out to under a) di erent price combinations (current and organic), b) di erent mechanisms for changing environmental concern (exogenous 'shock and decay', exogenous 'sustained incremental'). These simulation experiments were conducted against di erent levels of target ambition (UK Climate Change Committee's (CCC) core targets of % less dairy by and % by ; and extension targets of % by , along with the Eating Better alliance's more ambitious % by ). .
Key results showed that most model runs from scenarios with a dairy-PBM price relationship at today's levels, failed to deliver the core consumption targets (∼ % for and over % for . However, if all dairy milk were to be priced organically, this situation improves markedly with only around % of scenario model runs not meeting core and targets. Although both simulation experiments that explored environmental concern and milk choice showed larger decreases in dairy milk consumption, they produced two very di erent sets of results. Temporary concern 'shocks' did not structurally change consumption patterns, however, the introduction of small, but permanent, incremental gains to environmental concern and agent milk choice influence, actually resulted in dairy declining and falling beneath plant-based milks at around . .
For the more ambitious % targets, scenarios using a representation of today's price relationship did not produce a single successful model run. However, organic pricing was able to almost universally achieve the % target level within a reasonable band of tolerance. The study highlighted several areas for future research and data needs, including: more recent, frequent, and accessible empirical data on food influences; and ABM development of specific plant-based alternatives e.g. oat, soya, almond.
. This paper presented the first attempt at applying an agent-based model of food influence to future scenarios and trajectories of UK dairy and plant-based milk consumption, to achieve climate-related dietary reduction targets. From a policy perspective, successful scenarios for core CCC targets displayed plausible trajectories, and in this sense, o er supportive evidence by way of the feasibility of modelled rates-of-change. However, current price relationships between dairy and PBM may pose a barrier to achieving desired targets, and policy should consider measures to redress the imbalance of milk 'price' vs its social, economic and environmental 'cost'. To support widespread adoption of sustainable diets, the UK public should experience deliberative and meaningful environmental-related information and interaction to help engender sustained positive changes to environmental concern.
. Probability of interacting The probability of an agent interacting (exchanging information on milk choice function scores) with other agents in its network.
. Initial habit of incumbent The initial number of consecutive choices that have returned the same majority milk type.
. Social blindness The probability that an agent has the ability to perceive the impact of its choice and therefore the option of evaluating it.
. Post-choice justification The threshold beyond which an agent will simply justify the discrepancy between its values and behaviour (milk choice impacts), rather than act to resolve it.
. Cognitive dissonance threshold The threshold below which any discrepancy between an agent's values and its behaviour (milk choice impacts) will not trigger a state of cognitive dissonance.
. No. of neighbours The number of neighbours in an agent's network.
. Perception of health impact of PBM The perception of the health impact of PBM.
. Perception of environmental impact of PBM The perception of the environmental impact of PBM.
. Gradient of probability disposition The slope of the function that determines how quickly the probability of being disposed to consider choice of milk as a function of the informational entropy of milk choices in an agent's neighbour network.
. Perception of health impact of dairy The perception of the health impact of dairy milk.
. Perception of environmental impact of dairy The perception of the environmental impact of dairy milk.

Further calibration details
The calibration exercises used an (single and bi-objective) evolutionary algorithm (EA) implemented in Python (Van Rossum & Drake ). The NetLogo model and Python were linked via the NL PY package (Gunaratne ), which allows the remote control, execution, and analysis of the model from within a Python environment (in our case Jupyter). The DEAP Python package was used to execute the EA (Fortin et al. ). Specifically, we employed the "Mu Plus Lambda" algorithm, using a simulated binary crossover, polynomial bounded mutation and the NSGA-II selection algorithm. Candidate parameter sets were drawn from a uniform distribution over upper and lower bounds and an initial population of individual sets were created, with the algorithm running over generations.

Additional results
Figure shows sensitivity analysis for scenarios within the milk pricing simulation experiment (S -S ). Temporal ( -year time step, -) variance-based global sensitivity analysis was conducted for simulated PBM consumption. Sensitivities are a sum of first order values and those due to interactions between parameters. Across all values, the mean proportion of sensitivity due to interactions was %, with % due to parameters. Overall, scenarios S and S were relatively stable but S and S decreased over time. Further, S and S (note, results are for a % reduction by not target) show comparatively lower total variance, an unsurprising result, given the single, rather than bi-objective optimisation. In general, across scenarios, parameter sensitivity was fairly equal with no dominance of one parameter over another. For scenarios S -S , maximum sensitives were around the %-% range, and minimum were around %. For scenarios S -S , maximum sensitivity proportions were higher (within overall less variance) at %-% of variance. The most common parameter for minimum sensitivity was no. (Gradient of probability distribution), occurring for three scenarios (S , S , S ). The most common parameter for maximum sensitivity was no. (Perception of health impact of PBM), occurring for scenarios S and S . Figure shows a scenario exploration that considers a declining, rather than largely constant, total average milk consumption. Declining milk consumption is modelled as an annual decrease from levels, based on average rates of decline from to . By visual inspection, scenario S has a greater proportion of dairy (navy lines) model runs hitting the target level ( % reduction) than scenario S , although over a larger spread. For scenarios S and S , that both consider dairy milk price rising to organic levels, there does not appear to be a marked di erence in the proportion of successful runs (i.e., hitting the target). Declining consumption leads to lower adoption and consumption of plant-based milks, as a sizeable portion of the target is instead met with reduced demand. Indicatively, approximately % of the target level is due to less overall consumption and % due to substitution to plant-based alternatives. Figure : Additional scenarios (S and S ) that considered declining total consumption, at average rates from -, rather than stable total milk consumption (S and S ).
Figure shows the result of resampling di erent sets of , responses from both the ESS and BSA. Scenario S is selected for the resampling analysis, based on a single parameter set of the eight sets produced from calibration. This parameter set was selected on the basis of producing simulations runs that had the smallest mean summed absolute di erence with the arithmetic mean of all the simulation runs of all the calibrated parameter sets. Figure : Mean values of simulation outputs for scenario S with di erent populations of , agents, constructed with samples of BSA and ESS data.
The means and distribution of these results were compared at the point of the final time-step ( ). A one-way ANOVA test confirmed that the means were significantly di erent (F (9, 3070) = 107.12, p < 0.001). Post-hoc analysis to understand which groups were significantly di erent was conducted via Tukey's Honestly Significant Di erences (HSD) (α = 0.05).
, and the distributions of simulation outputs were analysed for stability. That is, we assessed at what point the change in variance from simulation runs under di erent sized populations slowed or stopped. This assessment is based on calculating the coe icient of variation (CV) for simulation means and standard deviation at the final timestep ( ) for each di erent population size. Figure b (right-hand side) shows that CV decreases as population size increases up to , agents, therea er, CV remains relatively stable at around . . Figure : Comparing the simulation output and coe icient of variation of scenario S under di erent agent population sizes.

Appendix B Model description -Additional details Emergence
The key results of modelled outputs that emerge from the behaviours and interactions of individuals are the macro-level average consumption of milk choice among the simulated population, the trajectories of these curves, and their proximity to delivering on dairy reduction target levels.

Interaction
Individuals interact with other individuals through a social network (small-world structure) where information exchange occurs. The mechanism by which information exchange occurs is not explicitly modelled, rather, this is governed by a probability of interaction and a social influence weighting

Stochasticity
Information on health and environmental impacts of di erent milk choice options that agents perceive is randomly drawn from a normal distribution with mean values. In the environmental concern simulation experiment, the occurrence of a public concern 'shock' is governed by a probability informed by observed data. Further, stochasticity is reflected in the logistic function of neighbour milk choice information entropy and probability of an agent becoming disposed to consider their alternatives.

Heterogeneity
Heterogeneity is represented by the assignment of state variables among the agents. Principally, agents have di erent milk choice influence weightings operationalised from British Social Attitude survey data, and di erent basic human values assigned according to the distribution of UK results data from the European Social Survey .

Observation (incl. Emergence)
At each time step, the component choice functions and decision-making function for each choice and each agent is collected.

Cognitive perception
Memory e ects are included, with agents able to store a limited amount of averaged information from a set number of previous time-steps. At each new time-step, new information is added to a rolling average of previous time-steps and information is removed beyond a threshold governed by the memory length parameter. The upper and lower memory bounds ( , ) follow the El Farol NetLogo model by Rand & Wilensky ( ).

Evaluation
Agents have the possibility to evaluate the choices they make. Evaluation o ers a mechanism for agents learn from prior experience and use this to inform future decisions. Cognitive dissonance in food choice is one theory of evaluation (see Ong et al. for review), and we model this through a conceptualisation of tension between an agent's human values and their milk choice behaviour. Here we use the theory of basic human values to assign two values (security and universalism) to each agent, reflecting, broadly, their position on health and their position on the environment. High importance for universalism values are associated with a deeper concern and action toward environmental issues (Schultz et al. ; Schwartz ). Within Schwartz's theory of basic human values, health is orientated to the security value (Schwartz ). The European Social Survey (ESS) includes questions from Schwartz basic human values. We take UK responses for universalism and security questions and operationalise them to give a distribution of values relating to the environment and health.
Each milk choice has an associated health and environmental impact. At each time step, the aggregate impact of the choices is calculated and then compared against the agent's values on a relative basis. If this relative impact is within a given proximity to their value position, determined by the 'cognitive dissonance threshold' parameter, no feedback is sent. However, if the di erence is su iciently large, the agent enters a state of cognitive dissonance whereby their actions are incongruent with the values they hold. Here, agents pursue the least costly path to try and escape this uncomfortable state. They will either reconsider their behaviour (next choice) and become spontaneously disposed and alter the weight (± % per time-step) they ascribe to health or environmental components versus physical (price) aspects, or they will alter their value base slightly (±% per time-step) to better fit the choices they make. If the di erence between impact and value base is too large, given by the 'justification' parameter, agents simply rationalise this dissonance and once again no feedback occurs.