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Thorndike’s law of effect in environmental policy

Thorndike’s law of effect in environmental policy


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For a long time, UK’s environmental policies have been heavily based on taxes and reliefs for businesses, as opposed to

bonusses (GOV.UK, 2014). As an interdisciplinary prime minister, I am aware of the work of the psychologist B.F. Skinner on

operant conditioning – learning that occurs based on the consequences of one’s action – which demonstrates that positive

reinforcement is the most effective way to strengthen the likeliness of a behaviour (Mcleod, 2018). In other words, rather than

punishing the companies with a fine when they do not comply with environmental policies, I should reward them when they do.

Description of the game:

Companies (player 1) can make the effort required for the environment (R) or not (r). The State (player 2) can audit the company

(C) or not (c). When each decide, even if not synchronous in time, they are not aware of each other’s choice. Therefore, the

game is a simultaneous move game. This model is sometimes called inspection game and often used to describe the problem of

verification of arms control and disarmament (Avenhaus, 2004), or insurance schemes (Lemaire, 1991). It goes as follow:

– for companies, respecting the policies involves a cost i, encompassing innovation and

implementation of environment-friendly technology (reducing soil and water pollution, or

emissions of greenhouse gases into the air, shrinking their waste stream…).

– for the UK environment agency, an audit costs a.

– if a company complies to environmental guidelines (we assume that an audit will never

fail to detect a fraud) the benefit for the environment is b. In that case, the UK

government gives out a reward w, as an incentive to encourage further effort.

– if the company does not comply with policies but it is not audited, the Environment

agency gives w by default (only if the State checks and finds out the company commits a

fraud, then it does not receive w).

Case study:

As prime minister, I am very proud of the global success of the British multinational

firm Tesco – the sixth biggest company in the UK by revenue in 2017

(, 2018), with over 3 400 stores and 440 000 people employed in

the UK (George, 2018) -. The retail giant is leader in Great-Britain with a 27.8 market

share of grocery stores (Statista, 2019). It seems to be the ideal test for my model.

Following the Paris Climate Agreement, the company aligned its targets with a 1.5

degree trajectory. Their commitment and their work with external experts to set

ambitious science-based objectives so as to limit climate change also demonstrate

their support for the UN’s Sustainable Development Goals (Umeasiegbu, 2017). Its

initiatives included switching the refrigerant gas used in 1″,200 of its UK stores to a

lower Global Warming Potential (GWP) gas and fostering behaviour change in

operating equipment in their stores and distribution centres. In 2016/17, it invested

in LED and fans efficiency too, and remarkably achieved 100 per cent renewable electricity in the UK and Ireland, supported by

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renewable certificates (Tesco plc, 2018). To achieve this, the company has invested more than £700m in energy and

refrigeration efficiency between 2007 and 2017 (Quinn, 2017), we will average this expense to i = 700m/10 = 70 million per year.

Furthermore, I assume that for a company each innovation is more complex than the previous one and thus more expensive, I

will assume by 10%, thus I will normalise w = 1.1i = 1.1 x 70m = £77m. In their 2016-2017 annual report, Tesco estimated its total

carbon footprint to 3 891 822 tonnes of carbon dioxide equivalent (Tesco PLC, 2018). It claimed to have reduced their net

carbon intensity per square foot of retail and distribution floor by 10% compared to the previous year. This corresponds to 0.1 x

3 891 822 = 38 9182.2 tonnes of CO2e emissions saved for an equal surface area compared to 2016 levels. The United States

Environmental Protection Agency (2017) measures the long-term damage done by a tonne of CO2 emissions in a given year, in

order to value the climate impacts of rulemakings. Estimates of this ‘social cost’ of carbon increase over time, from $105 in 2015

to $123 in 2020, as we expect future emissions to produce larger incremental damages as economic and physical systems

Players: N = {1″,2}

Actions: A1 = {R”,r} A2 = {c”,C}

A = A1 x A2 = (R”,C) , (r”,C) , (R”,c) , (r”,c)


u1(R”,C) = w-i

u1(r”,C) = 0

u1(R”,c) = w-i

u1(r”,c) = w

u2(R”,C) = b-w-a

u2(r”,C) = -a

u2(R”,c) = b-w

u2(r”,c) = -w


• On the one hand, the companies prefer to comply if an inspection is to

happen, and not to comply otherwise (to both save i and get w ‘for free’). If

the company complies, it is indifferent to whether the State decides to audit.

Therefore preferences for player 1 are (r”,c) ≻ (R”,C) = (R”,c) ≻ (r”,C).

• On the other hand, I prefer overall that the company complies with the

policies, as my goal is to maximise the benefit for the environment.

Moreover, I prefer to inspect if the company does not comply, to save w”,

that the company does not deserve. Nonetheless, I prefer not to inspect if it

follows the rules, so as to save the cost of the audit. This results in my

preferences (player 2) being the following: (c”,R) ≻ (C”,R) ≻ (C”,r) ≻ (c”,r).

become more stressed in response to climate change. As we focus on 2017, we can take the average (105+123)/2 = $114 =

£89.03 at the current rate. Thus b = 389 182.2 x 89″,03 = £34 648 891 rounded to the closest unit.

Last but not least, the cost of an environmental audit can vary from “$2″,500 for very simple, small facilities to well over $40″,000

for very large, complicated facilities” (Ruhlin, 2016), according to RMA Environmental services, which help companies to get into

compliance with environmental laws, rules, and regulations. They provide an exhaustive and sensible explanation of the

variables that impact audit costs & prices (size, type and number of facilities, the scope, who is involved, etc). Therefore, for

Tesco we will retain the figure a = $40 000 = £31 209.


• Players’ best responses are highlighted in their respective colours on the matrix. They do not intersect, which means that

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there is no pure strategy Nash equilibrium. Consequently, we shall check for mixed strategies.

• In a mixed strategy Nash equilibrium, players seek to maximize the expected utility of ui(a): E[ui(ai, a-i)] abreviated EUi . In

order to do this, each player has a behavioural strategy which is a probability assignment to each pure strategy. In our case”,

p1 = (pr, (1-p)R) and q2 = (qC, (1-q)c). Probability distribution over all actions must sum to 1.

• Player 1’s mixed strategy assumes his opponent is indifferent between his own pure strategies.

We assume 2’s Expected Utilities equal to each other (i.e. indifferent), we calculate both EUr and EUR given a function of

some mixed strategy of Player 1 and equate them. We then repeat algorithm for Player 2.

The blue line is q(p) -best response of player 1-, whereas the green line is p(q) -best

response of player 2:

• If p is below 0.0004, Player 2 chooses c i.e. q will equal 0.

• If p is 0.0004, q is any value, Player 2 is indifferent.

• If p is above 0.0004, Player 2 chooses C i.e. q will equal 1.

• If q is below 0.9, Player 1 chooses r i.e. p will equal 1.

• If q is 0.9, p is any value, Player 1 is indifferent.

• If q is above 0.9, Player 1 chooses R = p will equal 0.

Discussion and conclusion

The probability that the company will not comply with the policies is very low, insofar as the state audits almost all the time. If

the State fails to audit a company 90% of the time, an environment-friendly decision of the later would be deemed irrational.

Conversely, if it manages to send officers to check so regularly on the different companies, firms will make the effort virtually for

sure. My inner optimist tells me that it is good news, but she is wrong for two reasons. First, constant auditing, which price a will

be systematically paired with reward w, will come at a very high cost for the State, even though I bear in mind that fostering

R&D in sustainability will eventually benefit the businesses, as well as the State as an economic power. Secondly, what I wished

was not only to prevent pollution, but also to encourage spontaneous initiatives by the companies. Nosenzo et al. (2013), who

investigated ‘Bonuses versus Fines in Inspection Games’, also reached the conclusion that “the effect of bonuses on compliance

is much weaker than predicted”. I must point out that in this game the incentive for the companies to comply is extrinsic, but

energy efficiency improvement saves enormous internal running costs. For instance, Tesco cut its electricity bill by £200m a

year. Apart that it neglects this crucial factor, the issue with the model is that, in monetary value, pollution abatement costs are

much greater than the benefit for nature and society (i > b). This apparent ‘waste of money’ strongly justifies the temptation of

the company to deviate. However, the SC-CO2 estimate we used “includes changes in net agricultural productivity, human

health, property damages from increased flood risk, and changes in energy system costs. However, given current modelling and

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data limitations, it does not include all important damages” (United States Environmental Protection Agency, 2017). If firms do

not take upon themselves to address climate change now, the cost on the long-term will be immense. Luckily, when the US

President withdrew from COP21 Paris agreement, we have witnessed that firms did not perceive it as an opportunity to continue

causing damage the planet (Nogrady, 2017). On the contrary, it led to movements such as the open letter We Are Still In, a

declaration signed by political leaders and giants like Apple, Walmart, Microsoft, Adidas, Facebook and Google. This gives hope.


p(-77×106) + (1-p)(-42 351 109) = p(-31 209) (1-p)(-42 382 318)

-77x106p – 42 351 109 + 42 351 109p = -31 209p – 42 382 318 + 42 382 318p

-77x106p + 42 351 109p + 31 209 p – 42 382 318p = – 42 382 318 + 42 351 109

p (-77×106 + 42 351 109 + 31 209 – 42 382 318) = -31209

p = -31209/(-77×106)

p* = 0.0004


77×106(1-q) – 0q = 7×106(1-q) + 7x106q

77×106 – 77x106q = 7×106 – 7x106q + 7x106q

– 77q + 7q – 7q = – 77 + 7

-77q = -70

q = (-70) / (-77)

q* = 0.9

MSNE = {p* = 0.0004, q* = 0.9}


Avenhaus, R. (2004). Applications of inspection games. Mathematical Modelling and Analysis, 9(3), pp.179-192.

George, S. (2018). Tesco cuts carbon emissions 13% in a year. [online] Available at:–in-a-year/ [Accessed 9 Feb. 2019]. (2018). TOP 50 Companies in UK by Revenue in 2017. [online] Available at: [Accessed 11 Feb. 2019].

GOV.UK. (2014). Environmental taxes, reliefs and schemes for businesses. [online] Available at:

and-reliefs [Accessed 9 Feb. 2019].

Lemaire, J. (1991). Cooperative Game Theory and its Insurance Applications. ASTIN Bulletin, 21(01), pp.17-40.

Mcleod, S. (2018). B.F. Skinner | Operant Conditioning. [online] Available at: [Accessed 9 Feb. 2019].

Nogrady, B. (2017). No more business as usual: the corporates stepping up to save the planet. [online] the Guardian. Available at:

save-the-planet [Accessed 9 Feb. 2019].

Nosenzo, D., Offerman, T., Sefton, M. and van der Veen, A. (2013). Encouraging Compliance: Bonuses Versus Fines in Inspection

Games. Journal of Law, Economics, and Organization, 30(3), pp.623-648.

Quinn, I. (2017). Tesco sets itself ‘high ambition’ carbon reduction targets. [online] Available at:

targets/552789.article [Accessed 11 Feb. 2019].

Ruhlin, D. (2016). The true cost of an environmental audit. [online] Available at: [Accessed 10 Feb. 2019].

Statista. (2019). Great Britain: Grocery market share YoY 2013-2018 | UK Statistics. [online] Available at: [Accessed 10 Feb.


Tesco plc. (2018). Improving energy efficiency in our stores and distribution. [online] Available at: [Accessed 8 Feb.


Tesco PLC (2018). Tesco PLC Annual Report and Financial Statements 2017. [online] Welwyn Garden City, p.23. Available at: [Accessed 10 Feb. 2019].

Umeasiegbu, K. (2017). Tesco commits to use 100% renewable electricity by 2030. [online] Tesco plc. Available at: [Accessed 10 Feb. 2019].

United States Environmental Protection Agency (2017). The Social Cost of Carbon | Climate Change | US EPA. [online] Available at:

[Accessed 9 Feb. 2019].

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