The Wolfram Emerging Leaders Program: Boot Camp is a year-long program for talented high school students. Students spend a semester learning Wolfram Language, and then work in a small group over the second semester to do either a research-based project or a product development–based project under the guidance of an expert mentor. Projects are centered around a topic at the intersection of the group’s interests.
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With contributions from Vandit Rikhi
Introduction
Introduction
Background:
Background:
The Organization of the Petroleum Exporting Countries (OPEC) presents ongoing problem-and-resolution cycles of strategic coordination dependent on each member’s economic sovereignty. By collectively constraining crude-oil production, member states can reduce aggregate supply and balance favorable market prices per unit sold, thereby increasing output value retained across the group. The same arrangement can also create incentives for individual members to deviate from their agreement; if other producers abide by output restraint, an elect country may opt to increase short-term revenue by producing above its designated quota whilst continuing to benefit from the higher prices induced from the collective’s imposition on the products’ market values. Thus, OPEC’s success is dependent on appropriate abidement of rulings such that benefits become a consequence of the collective cooperation of the membership against the potential of individual incentives to undermine price action.
The tensions associated with OPEC compliance are a natural problem for game-theoretic analysis given expected equivalences sought for based on disparate, yet common motive players. In a non-cooperative setting, each producing nation chooses their output in response to the quantities entering the market of other members, affecting market pricing. A Nash Equilibrium of the situation identifies configurations in which no participant can improve its own pay-off through only unilateral changes - however self-serving - in strategy. Meanwhile, a Cournot framework provides a corresponding model in which the strategic choice would be a focus on the production quantity. Such an equilibrium does not need to maximize the combined welfare of all participants. Collectively preferable production agreements may remain vulnerable to unilateral deviation even when the collective benefits from the continued existence of cartelized trade economics.
Simple Prisoner’s Dilemma interpretations may not suffice to describe the OPEC situation, as member states’ incentives vary significantly across multiple metrics. These include production capacity, political agendas, cost of extraction, economic dependence on oil and gas revenue, government capacity to influence national industry, and spare capacity. For instance, a margin-impacting barrel produced by a smaller exporter may not necessarily have the same market consequence as a strategic production shift by Saudi Arabia. Therefore, incentives associated with quota compliance should be expected to differ across the members, disproportionate contributions that satisfy individual compromises and the collective agreement.
The tensions associated with OPEC compliance are a natural problem for game-theoretic analysis given expected equivalences sought for based on disparate, yet common motive players. In a non-cooperative setting, each producing nation chooses their output in response to the quantities entering the market of other members, affecting market pricing. A Nash Equilibrium of the situation identifies configurations in which no participant can improve its own pay-off through only unilateral changes - however self-serving - in strategy. Meanwhile, a Cournot framework provides a corresponding model in which the strategic choice would be a focus on the production quantity. Such an equilibrium does not need to maximize the combined welfare of all participants. Collectively preferable production agreements may remain vulnerable to unilateral deviation even when the collective benefits from the continued existence of cartelized trade economics.
Simple Prisoner’s Dilemma interpretations may not suffice to describe the OPEC situation, as member states’ incentives vary significantly across multiple metrics. These include production capacity, political agendas, cost of extraction, economic dependence on oil and gas revenue, government capacity to influence national industry, and spare capacity. For instance, a margin-impacting barrel produced by a smaller exporter may not necessarily have the same market consequence as a strategic production shift by Saudi Arabia. Therefore, incentives associated with quota compliance should be expected to differ across the members, disproportionate contributions that satisfy individual compromises and the collective agreement.
Purpose:
Purpose:
This study investigates the degree to which a Nash Equilibrium postulate and Cournot competition can reveal heterogenous incentives and raise insights on OPEC conformation. Elaborated: “To what extent can non-cooperative game theory explain differences in the production quota compliance of OPEC members, and under what conditions can cooperation persist despite incentives to deviate?” Here, a computational model is used to identify the strategic mechanisms that would make coordination of the cartelized system stable, or unstable.
Structure:
Structure:
Firstly, the data on historical production and quotas is used to characterize starting differences in compliance behaviours across OPEC members. The compliance is measured as the ratio of actual production to the production target, which allows each nation to be compared in terms of the average deviation, any variance, and behaviour changes over time. This stage establishes that OPEC members are not merely homogeneously strategizing actors, thereby providing a basis for more complex and descriptive models.
Secondly, static Cournot models are composed for investigating production incentives. A preliminary 2-member adversarial analysis is used, whose illustrative foundation first considers Iraq-Saudi Arabia discrepancies. This was chosen given apparent outlier features of Saudi Arabia that naturalized its status as premiere and enforcer of the OPEC alliance. From this outlier basis, it became possible to intuitively extend to an asymmetric, multi-country game theoretic approach. In this case, each producer is assigned a marginal production cost and a feasible production range. The production decisions would affect the oil price on the market and the demand specifications are analyzed according to benchmarking. The resulting quantities from non-cooperative scenarios and profits are then compared with quota-based outcomes, allowing to determine whether or not members have economic incentives to deviate from cooperation. This way, rather than assuming that deviation between members has the same meaning, it becomes possible to discuss both the direction and magnitude of the deviation as a function of each country’s very different characteristics.
Because the price component is primarily used for comparison, estimation of changes in aggregate output vs. producer payoffs is more seriously used against affected pricing. This is done relative to a common baseline. Historical Brent price observations are therefore taken as a shallow test rather than as evidence that the collective output alone determines market price.
Thirdly, the analysis studies repeated interactions, whereby OPEC members are postulated to not make production decisions in isolated circumstances, but instead, repeatedly observe each other’s behaviour. This brings in a renegotiation dynamic, and production arrangements can then be conducted to respond to previous compliance rules or non-compliance behaviour. Therefore, a simulation of repeated strategy can examine how behavioural rules, such as compliance, regular deviation or total defection, reciprocating actions (in response to other’s actions), trigger reasons, and enforcement by Saudi Arabia, can all affect long-term cooperation. Saudi Arabia is treated separately given its sheer scale, high spare capacity, and influence over aggregate supply - hence, its unique ‘enforcer’ position.
Secondly, static Cournot models are composed for investigating production incentives. A preliminary 2-member adversarial analysis is used, whose illustrative foundation first considers Iraq-Saudi Arabia discrepancies. This was chosen given apparent outlier features of Saudi Arabia that naturalized its status as premiere and enforcer of the OPEC alliance. From this outlier basis, it became possible to intuitively extend to an asymmetric, multi-country game theoretic approach. In this case, each producer is assigned a marginal production cost and a feasible production range. The production decisions would affect the oil price on the market and the demand specifications are analyzed according to benchmarking. The resulting quantities from non-cooperative scenarios and profits are then compared with quota-based outcomes, allowing to determine whether or not members have economic incentives to deviate from cooperation. This way, rather than assuming that deviation between members has the same meaning, it becomes possible to discuss both the direction and magnitude of the deviation as a function of each country’s very different characteristics.
Because the price component is primarily used for comparison, estimation of changes in aggregate output vs. producer payoffs is more seriously used against affected pricing. This is done relative to a common baseline. Historical Brent price observations are therefore taken as a shallow test rather than as evidence that the collective output alone determines market price.
Thirdly, the analysis studies repeated interactions, whereby OPEC members are postulated to not make production decisions in isolated circumstances, but instead, repeatedly observe each other’s behaviour. This brings in a renegotiation dynamic, and production arrangements can then be conducted to respond to previous compliance rules or non-compliance behaviour. Therefore, a simulation of repeated strategy can examine how behavioural rules, such as compliance, regular deviation or total defection, reciprocating actions (in response to other’s actions), trigger reasons, and enforcement by Saudi Arabia, can all affect long-term cooperation. Saudi Arabia is treated separately given its sheer scale, high spare capacity, and influence over aggregate supply - hence, its unique ‘enforcer’ position.
Summary of Process:
Summary of Process:
In summary, this step-by-step investigation answers two main concerns: (1) why deviation for a member may be profitable in a given production period; (2) why cooperation may survive despite member deviations in the long-term. This study presents that a Nash Equilibrium is useful for identifying the former concern, but on its own, cannot address the latter. Continued cartelized coordination is understandable as the outcome of various asymmetric incentives that still operate within a strategic relationship for the collective, such that short-term gains from any deviation can be weighed against subsequent reactions or response by other members.
This project aims to methodologically explain conventional circumstances rather than predict. By consulting historical compliance measures and asymmetric Cournot competitions, pay-off comparisons, and repeated strategic interaction. The experimental investigation is pursued in the Wolfram Language to provide a computational framework for examining the mechanisms that underlie OPEC compliance. Despite considering multiple market and institutional complexities, the model still finds that cooperation creates substantial value for the collective. This is the case even when distribution of such value is generated through varying incentives across the membership to comply, deviate, or enforce the alliance’s agreement.
This project aims to methodologically explain conventional circumstances rather than predict. By consulting historical compliance measures and asymmetric Cournot competitions, pay-off comparisons, and repeated strategic interaction. The experimental investigation is pursued in the Wolfram Language to provide a computational framework for examining the mechanisms that underlie OPEC compliance. Despite considering multiple market and institutional complexities, the model still finds that cooperation creates substantial value for the collective. This is the case even when distribution of such value is generated through varying incentives across the membership to comply, deviate, or enforce the alliance’s agreement.
Section 1 — Foundation: Data & 2-Player Game Theory
Section 1 — Foundation: Data & 2-Player Game Theory
Goal: To build and calibrate a foundational two-player Cournot model of Saudi Arabia and Iraq, using historical oil-market data to establish a baseline for analyzing quota compliance, overproduction and the Nash equilibrium between individual OPEC members.
1.1 OPEC Member & Quota Data Setup
1.1 OPEC Member & Quota Data Setup
Key data sources:
- OPEC Monthly Oil Market Reports (2017-2023)
- IEA Oil Market Reports
- Bloomberg commodity data (accessed via Wolfram FinancialData where available)
- OPEC Secretariat official quota tables
- OPEC Monthly Oil Market Reports (2017-2023)
- IEA Oil Market Reports
- Bloomberg commodity data (accessed via Wolfram FinancialData where available)
- OPEC Secretariat official quota tables
All production figures in thousands of barrels per day (kb/d)
All quota figures from official OPEC meeting decisions
Listing all OPEC Members as of 2023:
Note that Iran and Libya are frequently exempt from formal quotas due to
sanctions and civil conflict respectively; Venezuela produces far
below any plausible quota due to economic collapse.
I included them in member lists but flagged them as exempt where relevant.
All quota figures from official OPEC meeting decisions
Listing all OPEC Members as of 2023:
Note that Iran and Libya are frequently exempt from formal quotas due to
sanctions and civil conflict respectively; Venezuela produces far
below any plausible quota due to economic collapse.
I included them in member lists but flagged them as exempt where relevant.
In[]:=
opecAllMembers={"Saudi Arabia","Iraq","Iran","Kuwait","UAE","Venezuela","Libya","Algeria","Nigeria","Gabon","Congo","Equatorial Guinea","Cameroon"};
In[]:=
opecQuotaMembers={"Saudi Arabia","Iraq","Kuwait","UAE","Algeria","Nigeria","Gabon","Congo","Equatorial Guinea","Cameroon"};
Exempt members (sanctions, conflict, or economic collapse):
In[]:=
opecExemptMembers={"Iran","Libya","Venezuela"};analysisYears={2017,2018,2019,2020,2021,2022,2023};
Listing Assigned Quotas (thousands bbl/day, annual averages):
Sources: OPEC meeting decisions; 2020 values reflect the
historic April 2020 COVID-era cuts (9.7 Mb/d total cut)
Sources: OPEC meeting decisions; 2020 values reflect the
historic April 2020 COVID-era cuts (9.7 Mb/d total cut)
In[]:=
opecQuotaData=<|"Saudi Arabia"->{10058,10311,10311,8497,8497,10225,10000},"Iraq"->{4351,4651,4651,3590,3590,4431,4220},"Kuwait"->{2707,2707,2707,2092,2092,2809,2660},"UAE"->{2874,3068,3068,2590,2590,3168,3220},"Algeria"->{1089,1089,1089,914,914,1054,1000},"Nigeria"->{1685,1685,1685,1411,1411,1753,1590},"Gabon"->{197,202,202,175,175,186,180},"Congo"->{267,267,267,260,260,285,280},"Equatorial Guinea"->{130,140,140,102,102,119,110},"Cameroon"->{95,95,95,77,77,80,75}|>;
Listing Actual Production (thousands bbl/day, annual averages):
Sources: OPEC MOMR; IEA OMR; S&P Global Platts
Sources: OPEC MOMR; IEA OMR; S&P Global Platts
In[]:=
opecActualData=<|"Saudi Arabia"->{9955,10450,9780,9240,9150,10450,10100},"Iraq"->{4393,4652,4760,3912,4163,4535,4320},"Kuwait"->{2699,2731,2680,2127,2406,2706,2655},"UAE"->{2916,3082,3058,2682,2656,3063,3100},"Algeria"->{1041,1035,996,867,910,1010,980},"Nigeria"->{1533,1599,1780,1235,1545,1349,1420},"Gabon"->{210,218,213,174,186,197,190},"Congo"->{265,319,330,292,262,274,265},"Equatorial Guinea"->{148,139,120,91,107,106,95},"Cameroon"->{72,74,62,56,58,60,55}|>;
Exempt members - actual production only (no assigned quota):
In[]:=
opecExemptActualData=<|"Iran"->{3800,3530,2355,2050,2490,2550,3000},"Libya"->{820,1020,1130,380,1220,1380,1200},"Venezuela"->{1910,1351,877,558,637,724,770}|>;
Listing Approximate Marginal Production Costs (USD per barrel):
Sources: Rystad Energy, McKinsey, OPEC Secretariat estimates
These are rough breakeven costs, not full-cycle costs.
Sources: Rystad Energy, McKinsey, OPEC Secretariat estimates
These are rough breakeven costs, not full-cycle costs.
In[]:=
opecMarginalCosts=<|"Saudi Arabia"->3.5,"Iraq"->7.0,"Kuwait"->5.5,"UAE"->8.5,"Algeria"->24.0,"Nigeria"->22.0,"Gabon"->28.0,"Congo"->30.0,"Equatorial Guinea"->34.0,"Cameroon"->37.0,"Iran"->12.0,"Libya"->6.0,"Venezuela"->35.0|>;
Listing Historical Brent Crude benchmark prices (annual averages, USD/bbl):
In[]:=
brentPriceHistorical=<|2017->54.7,2018->71.3,2019->64.4,2020->41.7,2021->70.7,2022->99.0,2023->82.5|>
Out[]=
201754.7,201871.3,201964.4,202041.7,202170.7,202299.,202382.5
Extracting Historical Brent Crude Oil Prices:
In[]:=
brentPriceList=Values[brentPriceHistorical]
Out[]=
{54.7,71.3,64.4,41.7,70.7,99.,82.5}
Displaying OPEC Quota Members:
In[]:=
StringRiffle[opecQuotaMembers,", "]
Out[]=
Saudi Arabia, Iraq, Kuwait, UAE, Algeria, Nigeria, Gabon, Congo, Equatorial Guinea, Cameroon
Displaying OPEC Quota-Exempt Members:
In[]:=
StringRiffle[opecExemptMembers,", "]
Out[]=
Iran, Libya, Venezuela
1.2 Section 1 Deliverable: 2-Player Foundation Model
1.2 Section 1 Deliverable: 2-Player Foundation Model
Player 1 = Saudi Arabia (dominant cartel enforcer, low marginal cost)
Player 2 = Iraq (historically the highest-profile quota violator)
The Cournot duopoly gave me a baseline before I extended to N players.
Player 2 = Iraq (historically the highest-profile quota violator)
The Cournot duopoly gave me a baseline before I extended to N players.
Linear inverse demand: P(Q) = a - bQ
I calibrated a and b from historical data.
At Q = world_supply ~ 100,000 kb/d, P ~ 65 USD/bbl (approximate long-run avg)
Elasticity of demand for oil ~ -0.1 to -0.3 short-run.
Using midpoint: (dQ ÷ dP )× (P ÷ Q) ~ -0.2 => b = P ÷ (elasticity × Q)
b = 65 ÷ (0.2 × 100000) = 0.00325
a = P + bQ = 65 + (0.00325 × 100000) = 390
For the sub-game (Saudi + Iraq only, treating rest as fixed background):
Background supply = opecTotal - SA - Iraq + nonOPEC ~ 75000 kb/d
Residual demand for SA + Iraq: P = (390 - 0.00325 × 75000) - 0.00325 × (qSA + qIraq)
= 146.25 - 0.00325 × (qSA + qIraq)
So: aresidual = 146.25, bresidual = 0.00325
I calibrated a and b from historical data.
At Q = world_supply ~ 100,000 kb/d, P ~ 65 USD/bbl (approximate long-run avg)
Elasticity of demand for oil ~ -0.1 to -0.3 short-run.
Using midpoint: (dQ ÷ dP )× (P ÷ Q) ~ -0.2 => b = P ÷ (elasticity × Q)
b = 65 ÷ (0.2 × 100000) = 0.00325
a = P + bQ = 65 + (0.00325 × 100000) = 390
For the sub-game (Saudi + Iraq only, treating rest as fixed background):
Background supply = opecTotal - SA - Iraq + nonOPEC ~ 75000 kb/d
Residual demand for SA + Iraq: P = (390 - 0.00325 × 75000) - 0.00325 × (qSA + qIraq)
= 146.25 - 0.00325 × (qSA + qIraq)
So: aresidual = 146.25, bresidual = 0.00325
General 2-player Cournot Nash equilibrium:
In[]:=
cournotNashEquilibrium2[a_,b_,c1_,c2_]:=Module[{q1star,q2star,pstar,pi1,pi2},q1star=(a-2*c1+c2)/(3*b);q2star=(a-2*c2+c1)/(3*b);pstar=a-b*(q1star+q2star);pi1=(pstar-c1)*q1star;pi2=(pstar-c2)*q2star;<|"q1SaudiArabia"->q1star,"q2Iraq"->q2star,"TotalProduction"->q1star+q2star,"EquilPrice"->pstar,"ProfitSA"->pi1,"ProfitIraq"->pi2,"TotalProfit"->pi1+pi2|>];
Calibrated parameters for the Saudi-Iraq sub-market:
2-Player Cournot Nash Equilibrium (Saudi Arabia vs. Iraq):
Compare Nash quantities to assigned quotas (2023 values, in kb/d):
Key insight: The Cournot Nash production is higher than the quota,
confirming that without cartel enforcement, countries have incentive to overproduce.
confirming that without cartel enforcement, countries have incentive to overproduce.
Strategies: C = Comply (produce at quota), D = Defect (overproduce)
Payoff function for the 2-player cooperation game:
When a country defects, it produces at the Cournot Nash quantity
rather than the assigned quota. 2023 data has been used here.
When a country defects, it produces at the Cournot Nash quantity
rather than the assigned quota. 2023 data has been used here.
Build displayable grid:
Nash Equilibrium identification:
If Iraq plays C: SA prefers
If Iraq plays D: SA prefers
If SA plays C: Iraq prefers
If SA plays D: Iraq prefers
Both countries have a dominant strategy to defect, producing a (D,D) Nash equilibrium. This helps
explain why sustained quota compliance may require credible monitoring and penalties within OPEC.
explain why sustained quota compliance may require credible monitoring and penalties within OPEC.
Section 2 — Multi-Country OPEC Compliance Analysis & Cournot Model
Section 2 — Multi-Country OPEC Compliance Analysis & Cournot Model
Goal: Scale my model from the 2-player foundation to all 10 quota-bearing
OPEC members. I analyzed:
(a) Historical quota compliance ratios per country (2017-2023)
(b) N-player asymmetric Cournot equilibrium
(c) Multi-country payoff matrices
(d) Nash equilibrium in the N-player compliance game
(e) Price prediction validation against Brent crude history
(f) Statistical profiling of country compliance behavior
OPEC members. I analyzed:
(a) Historical quota compliance ratios per country (2017-2023)
(b) N-player asymmetric Cournot equilibrium
(c) Multi-country payoff matrices
(d) Nash equilibrium in the N-player compliance game
(e) Price prediction validation against Brent crude history
(f) Statistical profiling of country compliance behavior
2.1 Compliance Ratio Analysis (All Quota Members, 2017-2023)
2.1 Compliance Ratio Analysis (All Quota Members, 2017-2023)
Compliance Ratio = Actual Production ÷ Assigned Quota
ratio < 1.0 => Under-produced (over-compliant, delivered extra cuts)
ratio = 1.0 => Exactly at quota (perfectly compliant)
ratio > 1.0 => Over-produced (non-compliant, defecting from cartel)
ratio = 1.0 => Exactly at quota (perfectly compliant)
ratio > 1.0 => Over-produced (non-compliant, defecting from cartel)
Compute compliance ratio for each member-year:
Round for display:
Compliance classification:
Color coding for visualization:
Display Compliance Ratios by Country and Year (Actual/Quota):
Summary statistics per country:
Visualization 1: Compliance Heat Map (2017-2023)
Heat map of compliance ratios:
Heat map of compliance ratios:
Visualization 2: Per-country compliance time series
Visualization 3: Bar chart of mean compliance by country
2.2 N-Player Asymmetric Cournot Model (All OPEC Quota Members)
2.2 N-Player Asymmetric Cournot Model (All OPEC Quota Members)
I extended to an N-player Cournot model where each OPEC quota member
is a strategic player choosing a production quantity q_i.
Market setup:
- Linear inverse demand: P(Q) = A - BQ where Q = sum of all q_i + background
- Background supply (non-OPEC + exempt OPEC) treated as fixed
- Each country (i) maximizes profit: pi_i = (P(Q) - c_i) × q_i
Nash Equilibrium (best-response functions simultaneously satisfied):
FOC for country i: A - B × (Q - q_i) - B × (q_i - c_i) = 0
=> (A - B) × (Q - B) × (q_i - c_i) = 0 (wrong)
Correct FOC (Cournot):
d ÷ dq_i [(A - B × (q_i + Q_{-i}) - c_i) × q_i] = 0
A - B × Q_{-i} - 2 × B × q_i - c_i = 0
q_i = (A - c_i - B × Q_{-i}) ÷ (2B)
System of N equations solved simultaneously.
is a strategic player choosing a production quantity q_i.
Market setup:
- Linear inverse demand: P(Q) = A - BQ where Q = sum of all q_i + background
- Background supply (non-OPEC + exempt OPEC) treated as fixed
- Each country (i) maximizes profit: pi_i = (P(Q) - c_i) × q_i
Nash Equilibrium (best-response functions simultaneously satisfied):
FOC for country i: A - B × (Q - q_i) - B × (q_i - c_i) = 0
=> (A - B) × (Q - B) × (q_i - c_i) = 0 (wrong)
Correct FOC (Cournot):
d ÷ dq_i [(A - B × (q_i + Q_{-i}) - c_i) × q_i] = 0
A - B × Q_{-i} - 2 × B × q_i - c_i = 0
q_i = (A - c_i - B × Q_{-i}) ÷ (2B)
System of N equations solved simultaneously.
Parameters calibrated to global oil market (2023 reference year)
World oil demand ~ 100,000 kb/d at ~82.5 USD/bbl
Non-OPEC + exempt OPEC supply (treated as fixed background)
US ~13000, Russia ~10500, Canada ~4700, Brazil ~3500, ... + exempt OPEC
World oil demand ~ 100,000 kb/d at ~82.5 USD/bbl
Non-OPEC + exempt OPEC supply (treated as fixed background)
US ~13000, Russia ~10500, Canada ~4700, Brazil ~3500, ... + exempt OPEC
Iran 3000 + Libya 1200 + Venezuela 770 = 4970
Total world demand at 2023 price:
Residual demand for OPEC quota members:
Extract marginal costs for quota members (ordered list):
N-Player Cournot Nash Equilibrium Solver:
Solves the system of N best-response equations simultaneously
Solves the system of N best-response equations simultaneously
N-Player Cournot Nash Equilibrium (2023 Calibration):
Actual 2023 Brent Price: 82.5 USD/bbl
Key finding: Cournot NE quantities > quotas for most members, confirming the cartel reduces output below the non-cooperative equilibrium.
Key finding: Cournot NE quantities > quotas for most members, confirming the cartel reduces output below the non-cooperative equilibrium.
Cartel Cooperative Solution (Quota-Constrained maximization):
The OPEC cartel tries to maximize joint profit by restricting output. I solved for the joint-profit-maximizing quantity for each member, subject to total output = OPEC quota target. OPEC sets a total production target; then allocates quotas to members. I modeled the allocation that maximizes joint profit for a given price.
Total quota for 2023 (sum of individual quotas):
Cooperative price at total quota:
The elevated cooperative price should not be interpreted as a realistic forecast of the global oil price. It results from the simplifying assumption that changes in production by the two modelled OPEC members translate directly into changes in the global oil price, without fully accounting for responses from other producers. The value is therefore used primarily to illustrate the strategic interaction between Saudi Arabia and Iraq rather than to predict an exact market price.
The elevated cooperative price should not be interpreted as a realistic forecast of the global oil price. It results from the simplifying assumption that changes in production by the two modelled OPEC members translate directly into changes in the global oil price, without fully accounting for responses from other producers. The value is therefore used primarily to illustrate the strategic interaction between Saudi Arabia and Iraq rather than to predict an exact market price.
Cooperative profits at quota-implied price:
Compare cooperative vs. Cournot profits:
This surplus is what the cartel arrangement creates and why members have incentive to enforce it.
Best-Response Function Visualization:
For each member, plot best-response qty as a function of others’ total
Example: Saudi Arabia’s best response to other members’ aggregate production
For each member, plot best-response qty as a function of others’ total
Example: Saudi Arabia’s best response to other members’ aggregate production
Range: others produce 0 to 40000 kb/d
2.3 Multi-Country Payoff Matrix Analysis
2.3 Multi-Country Payoff Matrix Analysis
I constructed payoff matrices for all pairs of OPEC quota members, and then generalized to the full N-player compliance game.
Each country chooses: Comply (C) = produce at quota, or Defect (D) = overproduce. When a country defects, I model it as producing at its individual Cournot NE quantity.
The multi-player payoff matrix reveals:
- Which bilateral pairs have dominant defection strategies
- Which pairs might sustain cooperation under certain conditions
- The overall equilibrium of the N-player game
- Which bilateral pairs have dominant defection strategies
- Which pairs might sustain cooperation under certain conditions
- The overall equilibrium of the N-player game
Strategy profile: a list of “C” or “D” for each quota member, in the order of opecQuotaMembers. Quota quantities (2023) and Cournot NE quantities are used.
Price given a strategy profile:
Payoff to member i under profile:
2-Player Sub-Matrices (All Pairs):
For a pair (i, j), hold all others at Comply, compute 2x2 payoff matrix
For a pair (i, j), hold all others at Comply, compute 2x2 payoff matrix
Show 3 most interesting pairs: SA-Iraq, SA-UAE, Iraq-Nigeria:
N-Player Compliance Game: Full Strategy Space Analysis:
For the full 10-player game, I identified the Nash Equilibrium by checking whether any player has incentive to deviate from (C,...,C).
Key theorem: (D, D, ..., D) is always a Nash Equilibrium in the Cournot game. But (C, C, ..., C) is not a Nash Equilibrium unless side payments exist. I computed for each member the gain from unilateral defection while all others comply. If gain > 0, that member has incentive to deviate.
Payoff under full compliance:
Unilateral defection gain:
N-Player Game - Incentive to Deviate from Full Compliance:
(Positive gain = member benefits from defecting while others comply)
(Positive gain = member benefits from defecting while others comply)
Nash Equilibrium of full N-player game:
2.4 Historical Price Validation
2.4 Historical Price Validation
I tested whether my Cournot model can retrodict historical Brent crude prices by plugging in actual production levels. The model's predicted price is:
P_pred = A - B × (Q_total_OPEC_quota_members + adjustment)
P_pred = A - B × (Q_total_OPEC_quota_members + adjustment)
I also examined the relationship between OPEC compliance and price movements.
For each year, predict price from actual OPEC quota-member production:
Note: This is a simplified prediction. The model underpredicts absolute price level because it ignores non-OPEC supply variations, demand shocks, and speculation. I focused on relative changes / trends.
Correlation between OPEC total compliance and price changes:
Correlation: lower compliance (over-production) -> lower price?
Price change prediction: deltaP from deltaQ
When OPEC defects by 1000 kb/d, price falls by B × 1000
When OPEC defects by 1000 kb/d, price falls by B × 1000
Year-over-year comparison:
Predicted vs. Actual Brent Price (Simplified Cournot Model)
Predicted vs. Actual Brent Price (Simplified Cournot Model)
Note: Model error reflects demand-side factors, non-OPEC supply, and financial market dynamics not captured by the model used.
Regression - Actual OPEC production vs. Brent price:
Note: With only 7 data points, R-sq is limited. 2020 COVID outlier strongly
affects the fit; further analysis with quarterly data would improve this.
affects the fit; further analysis with quarterly data would improve this.
Visualization - Scatter plot with regression line:
2.5 Country Compliance Profiles & Statistical Analysis
2.5 Country Compliance Profiles & Statistical Analysis
I statistically clustered OPEC members by their compliance behavior, identifying:
- Reliable cooperators (consistently below quota)
- Systematic defectors (consistently above quota)
- Volatile members (high variance, context-dependent behavior)
This informed the strategy space for my Section 3 agentic simulation.
- Reliable cooperators (consistently below quota)
- Systematic defectors (consistently above quota)
- Volatile members (high variance, context-dependent behavior)
This informed the strategy space for my Section 3 agentic simulation.
Compliance features for clustering:
K-Means clustering: 3 clusters (cooperators, defectors, volatile)
Country Compliance Clusters:
Cluster scatter plot:
Quantitative country profiles:
Time-trend analysis: is compliance improving or worsening?
Compliance Trend (Linear Slope Over 2017-2023):
Positive slope = becoming more compliant over time
Positive slope = becoming more compliant over time
Geopolitical Event Overlay on Compliance:
Major geopolitical events affecting OPEC compliance
Major geopolitical events affecting OPEC compliance
Visualize compliance time series with geopolitical annotations:
2.6 Section 2 Summary & Findings
2.6 Section 2 Summary & Findings
Summary of Section 2 findings:
1. COMPLIANCE HETEROGENEITY: OPEC members exhibit very different compliance
profiles. Cameroon and Algeria are systematically under-quota (cooperative),
while Iraq and Congo frequently overproduce. Saudi Arabia is near-compliant
most years, leveraging its role as swing producer.
2. ASYMMETRIC COMPLIANCE INCENTIVES: The pairwise Cournot payoff matrices do not
reveal a uniform N-player Prisoner’s Dilemma. Saudi Arabia has a dominant incentive to
comply in the analyzed games, while Iraq, the UAE and Nigeria often have stronger
incentives to defect. This suggests a burden-sharing problem in which Saudi Arabia
absorbs more of the cost of restricting supply while some smaller producers benefit
from exceeding their quotas. Cooperation may therefore depend on repeated
interaction, credible monitoring and country-specific production constraints.
3. PRICE SENSITIVITY: The model estimates ~$0.12 price reduction per additional
100 kb/d of overproduction by OPEC quota members. The aggregate effect is
meaningful: 500 kb/d total overproduction -> ~$0.60/bbl price reduction.
4. MODEL LIMITATIONS:
- Simple linear demand ignores non-linearities at price extremes
- Non-OPEC supply treated as fixed (clearly wrong for US shale)
- No intertemporal optimization (countries are myopic in this model)
- Financial speculation and inventory effects not captured
1. COMPLIANCE HETEROGENEITY: OPEC members exhibit very different compliance
profiles. Cameroon and Algeria are systematically under-quota (cooperative),
while Iraq and Congo frequently overproduce. Saudi Arabia is near-compliant
most years, leveraging its role as swing producer.
2. ASYMMETRIC COMPLIANCE INCENTIVES: The pairwise Cournot payoff matrices do not
reveal a uniform N-player Prisoner’s Dilemma. Saudi Arabia has a dominant incentive to
comply in the analyzed games, while Iraq, the UAE and Nigeria often have stronger
incentives to defect. This suggests a burden-sharing problem in which Saudi Arabia
absorbs more of the cost of restricting supply while some smaller producers benefit
from exceeding their quotas. Cooperation may therefore depend on repeated
interaction, credible monitoring and country-specific production constraints.
3. PRICE SENSITIVITY: The model estimates ~$0.12 price reduction per additional
100 kb/d of overproduction by OPEC quota members. The aggregate effect is
meaningful: 500 kb/d total overproduction -> ~$0.60/bbl price reduction.
4. MODEL LIMITATIONS:
- Simple linear demand ignores non-linearities at price extremes
- Non-OPEC supply treated as fixed (clearly wrong for US shale)
- No intertemporal optimization (countries are myopic in this model)
- Financial speculation and inventory effects not captured
Section 3 — Repeated Strategy Simulation of OPEC Compliance
Section 3 — Repeated Strategy Simulation of OPEC Compliance
Section 3 Goal: Build a multi-period repeated strategy simulation with OPEC quota members being used as agents.
I modeled:
- Multiple strategy types (compliance strategies borrowed from game theory literature)
- Repeated game dynamics with reputation effects
- Saudi Arabia as a special enforcement agent
- External shocks (demand collapses, sanctions) as simulation parameters
- Multiple strategy types (compliance strategies borrowed from game theory literature)
- Repeated game dynamics with reputation effects
- Saudi Arabia as a special enforcement agent
- External shocks (demand collapses, sanctions) as simulation parameters
3.3 Baseline Simulation: Heterogeneous Strategies
3.3 Baseline Simulation: Heterogeneous Strategies
I ran the baseline simulation with a historically-motivated strategy assignment:
- Saudi Arabia: SaudiEnforcer (largest member, acts as cartel policeman)
- Iraq: AlwaysDefect (historically high non-compliance)
- Kuwait, UAE: TitForTat (respond to group behavior)
- Nigeria: Threshold (produces based on own profit performance)
- Algeria: GrimTrigger (strict but cooperative)
- Gabon, Congo, EqGuinea, Cameroon: AlwaysComply (small members comply to maintain membership)
Strategy assignment (order matches opecQuotaMembers)
{“Saudi Arabia”, “Iraq”, “Kuwait”, “UAE”, “Algeria”,
“Nigeria”, “Gabon”, “Congo”, “Equatorial Guinea”, “Cameroon”}
- Saudi Arabia: SaudiEnforcer (largest member, acts as cartel policeman)
- Iraq: AlwaysDefect (historically high non-compliance)
- Kuwait, UAE: TitForTat (respond to group behavior)
- Nigeria: Threshold (produces based on own profit performance)
- Algeria: GrimTrigger (strict but cooperative)
- Gabon, Congo, EqGuinea, Cameroon: AlwaysComply (small members comply to maintain membership)
Strategy assignment (order matches opecQuotaMembers)
{“Saudi Arabia”, “Iraq”, “Kuwait”, “UAE”, “Algeria”,
“Nigeria”, “Gabon”, “Congo”, “Equatorial Guinea”, “Cameroon”}
Run baseline simulation (48 rounds = 4 years, quarterly):
Extract time series:
Cumulative profits:
Summary statistics:
Per-agent final stats:
Visualizations:
Plot 1: Price over time
Plot 1: Price over time
Plot 2: Compliance rate over time
Plot 3: Cumulative profits by country
Strategy Comparison: All-Comply vs. All-Defect vs. Baseline
Run all-comply scenario:
Run all-comply scenario:
Run all-defect scenario:
Extract price histories:
Comparison plot:
Strategy Regime Comparison:
3.4 Sensitivity Analysis & Enforcement Scenarios
3.4 Sensitivity Analysis & Enforcement Scenarios
I tested how the simulation results change under different conditions, including demand shocks and stronger penalties for quota non-compliance.
Simulate a demand shock in rounds 10-14 (like 2020 COVID crash):
Overlay- baseline vs. shock:
Enforcement Strength Parameter Sweep:
Test what fraction of defectors triggers Saudi punishment
Modify SaudiEnforcer threshold- trigger after k defectors
Test what fraction of defectors triggers Saudi punishment
Modify SaudiEnforcer threshold- trigger after k defectors
Enforcement Threshold Sweep:
Conclusion
Conclusion
This study examined OPEC quota compliance across member states as an asymmetric strategic coordination and individuals’ alignment study. The central question was whether the incentives facing individual participants sustained compliance towards the collective’s production policies. Each country’s ability to alter its output in response to internal and/or observed external factors determines alignment with, or deviation from, group coordination efforts. Historical compliance analysis, static Cournot modelling, and repeated-strategy simulation were used to investigate the asymmetric dimensions of that problem.
Analysis Models:
Review of the historical data between OPEC member states showed starkly different behaviours between member states in terms of their adherence to production quotas: Iraq, Congo and Gabon all recorded average production as above their quotas, Algeria and Nigeria generally produced below their assigned quotas, whereas the United Arab Emirates showed nearly perfect compliance. Saudi Arabia was shown to have been generally close to its quota, but its compliance was prone to changing in different market conditions. This thus provides evidence for different members being given different characteristics in the final repeated-strategy simulation.
The static game-theoretic analyses reinforced the conclusion that quota adherence behaviours would vary significantly across the participants as a function of various metrics, resulting in disproportions that generated unique reasons for surprisingly common decisions (i.e., total compliance, short-term deviation, or total defection). In the simplified Saudi Arabia-Iraq case, non-cooperative production choices illustrate that there is a basic tension between a single member’s output and what is negotiated as the tempered, collective production. By extending scenarios to consider the entire list of OPEC asymmetric producers, however, the Saudi-Iraq case does not advise on a similar strategic response. The membership’s differences in costing, production capacity, politics, market influence, and more all result in complex incentive variations that create differing magnitudes and directions of deviation. As has been exhibited, for some producers, an additional output may be attractive whilst others continue to abide by the alliance’s agreement; however, for a sufficiently influential producer like Saudi Arabia, maintenance of the agreed restraint may actually generate greater pay-offs due to a disproportionately held share of price influence.
The described asymmetries provide a more useful interpretation of OPEC as opposed to a simple Prisoner’s Dilemma, which can oversimplify the individual equation. Production agreements create collective value by restricting aggregate supply, but the distribution of production cuts and surpluses can result in a burden sharing issue. Essentially, the stability of the arrangement would depend on whether the allocation of quotas across membership, and individuals’ own consequences of deviation, are sufficient to prevent solo state exploitation of other members’ exercised restraint.
The repeated-strategy analysis goes further to address a feature absent from a 1-period Nash Equilibrium, which is the repeated interaction of OPEC members. For instance, a producer contemplating deviation should consider not only its immediate benefits from production, but also the consequential responses of the other members. Reciprocal behaviours, trigger tactics, and reputational impacts are dynamic metrics that would present themselves. Furthermore, the enforcement by a large, restraint-tolerant producer like Saudi Arabia can influence pay-offs that would be associated with short-term deviation. It is important to note that the simulations do not necessarily predict the behaviour of particular member states, but they at least demonstrate that repeated interaction may sustain patterns that would otherwise be difficult to explain through the 1-shot scenario, alone.
Market Price Review:
Resulting market prices should be interpreted within a limited sense reflective of the simplicity of the models employed. The calibrated relationship is generally interesting for estimating direction and magnitude of price, as well as resulting profits; however, it cannot forecast real-world scenarios and is not intended to, especially when considering Brent crude values. Historical oil prices do reflect changes in global demand, non-OPEC production (external-global capacity influences), inventories during given trade epochs, individual financial constraints, or even geopolitical turmoil. These factors are beyond the scope of the simplified Cournot framework employed here. The empirical price analysis, therefore, serves as a starting diagnostic exercise which compares within a “limited influencers” criteria. It is not developed in the ways necessary to constitute causal validations of strategic plays.
Afterword:
The principal conclusion is that Nash Equilibria can explain important components of OPEC behaviour, but may be incomplete. The equilibrium used thus far identifies incentives that can make a negotiated production allocation across the collective unstable, and asymmetric multi-country analysis explains why incentives need not be identical across the membership. Repeated-game mechanisms provide a plausible framework for understanding how cooperation can persist despite the existence of profitable short-term deviations.
Most importantly, it can be observed that OPEC is not a truly disciplined cartel, may not continue on to be, and neither is it inevitably unstable. The repeated strategy institution that it forms constitutes producers with unequal costs, capacities, and market influences (including adjacent industry effects and politics). These effectors continually balance the value of collective restraint against individual members’ incentives to alter their production outputs. The Wolfram Language examination here demonstrates that game theoretic interpretations can be used to more aptly define the strategic tensions inherent, and further sampling approaches can make them even more measurable. This presents opportunity for further research and testing.
Future Work:
Building on the work done in this project, it is possible to strengthen the analytical framework through higher frequency quota and production data, including higher resolution inputs to improve pattern recognitions. Time-varying production capacity, improved estimation of residual oil demand, and a study of truly adaptive strategy selection would create more insightful and holistic approximations of behavioural dynamics and reactions. These add-ons would not alter the purpose of the model but would improve its analytical structure to divulge deeper insights into cooperation play vs. deviation energetics.
Analysis Models:
Review of the historical data between OPEC member states showed starkly different behaviours between member states in terms of their adherence to production quotas: Iraq, Congo and Gabon all recorded average production as above their quotas, Algeria and Nigeria generally produced below their assigned quotas, whereas the United Arab Emirates showed nearly perfect compliance. Saudi Arabia was shown to have been generally close to its quota, but its compliance was prone to changing in different market conditions. This thus provides evidence for different members being given different characteristics in the final repeated-strategy simulation.
The static game-theoretic analyses reinforced the conclusion that quota adherence behaviours would vary significantly across the participants as a function of various metrics, resulting in disproportions that generated unique reasons for surprisingly common decisions (i.e., total compliance, short-term deviation, or total defection). In the simplified Saudi Arabia-Iraq case, non-cooperative production choices illustrate that there is a basic tension between a single member’s output and what is negotiated as the tempered, collective production. By extending scenarios to consider the entire list of OPEC asymmetric producers, however, the Saudi-Iraq case does not advise on a similar strategic response. The membership’s differences in costing, production capacity, politics, market influence, and more all result in complex incentive variations that create differing magnitudes and directions of deviation. As has been exhibited, for some producers, an additional output may be attractive whilst others continue to abide by the alliance’s agreement; however, for a sufficiently influential producer like Saudi Arabia, maintenance of the agreed restraint may actually generate greater pay-offs due to a disproportionately held share of price influence.
The described asymmetries provide a more useful interpretation of OPEC as opposed to a simple Prisoner’s Dilemma, which can oversimplify the individual equation. Production agreements create collective value by restricting aggregate supply, but the distribution of production cuts and surpluses can result in a burden sharing issue. Essentially, the stability of the arrangement would depend on whether the allocation of quotas across membership, and individuals’ own consequences of deviation, are sufficient to prevent solo state exploitation of other members’ exercised restraint.
The repeated-strategy analysis goes further to address a feature absent from a 1-period Nash Equilibrium, which is the repeated interaction of OPEC members. For instance, a producer contemplating deviation should consider not only its immediate benefits from production, but also the consequential responses of the other members. Reciprocal behaviours, trigger tactics, and reputational impacts are dynamic metrics that would present themselves. Furthermore, the enforcement by a large, restraint-tolerant producer like Saudi Arabia can influence pay-offs that would be associated with short-term deviation. It is important to note that the simulations do not necessarily predict the behaviour of particular member states, but they at least demonstrate that repeated interaction may sustain patterns that would otherwise be difficult to explain through the 1-shot scenario, alone.
Market Price Review:
Resulting market prices should be interpreted within a limited sense reflective of the simplicity of the models employed. The calibrated relationship is generally interesting for estimating direction and magnitude of price, as well as resulting profits; however, it cannot forecast real-world scenarios and is not intended to, especially when considering Brent crude values. Historical oil prices do reflect changes in global demand, non-OPEC production (external-global capacity influences), inventories during given trade epochs, individual financial constraints, or even geopolitical turmoil. These factors are beyond the scope of the simplified Cournot framework employed here. The empirical price analysis, therefore, serves as a starting diagnostic exercise which compares within a “limited influencers” criteria. It is not developed in the ways necessary to constitute causal validations of strategic plays.
Afterword:
The principal conclusion is that Nash Equilibria can explain important components of OPEC behaviour, but may be incomplete. The equilibrium used thus far identifies incentives that can make a negotiated production allocation across the collective unstable, and asymmetric multi-country analysis explains why incentives need not be identical across the membership. Repeated-game mechanisms provide a plausible framework for understanding how cooperation can persist despite the existence of profitable short-term deviations.
Most importantly, it can be observed that OPEC is not a truly disciplined cartel, may not continue on to be, and neither is it inevitably unstable. The repeated strategy institution that it forms constitutes producers with unequal costs, capacities, and market influences (including adjacent industry effects and politics). These effectors continually balance the value of collective restraint against individual members’ incentives to alter their production outputs. The Wolfram Language examination here demonstrates that game theoretic interpretations can be used to more aptly define the strategic tensions inherent, and further sampling approaches can make them even more measurable. This presents opportunity for further research and testing.
Future Work:
Building on the work done in this project, it is possible to strengthen the analytical framework through higher frequency quota and production data, including higher resolution inputs to improve pattern recognitions. Time-varying production capacity, improved estimation of residual oil demand, and a study of truly adaptive strategy selection would create more insightful and holistic approximations of behavioural dynamics and reactions. These add-ons would not alter the purpose of the model but would improve its analytical structure to divulge deeper insights into cooperation play vs. deviation energetics.
Acknowledgements
Acknowledgements
I would like to thank Henry Gustafson for being an amazing mentor and always being available to help with anything I needed during the research process from debugging code to reviewing my minimum viable product. I would also like to thank Rory Foulger and Eryn Gillan for giving me the opportunity to be a part of this program and for their continuous support.
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Modeling OPEC quota compliance through game theory and Nash equilibrium analysis
by Ali Masomzadeh and Vandit Rikhi
Wolfram Community, STAFF PICKS, September 14, 2026
https://community.wolfram.com/t/28005
by Ali Masomzadeh and Vandit Rikhi
Wolfram Community, STAFF PICKS, September 14, 2026
https://community.wolfram.com/t/28005