A game theoretic approach the European
sanctions on Russia.
Is the withdrawal of sanctions part of a rational sanctions regime.
Bachelor Thesis
Economie en Bedrijfskunde Student: Luuk Mocking Student Number: 5927110 Supervisor: Ron van Maurik
Statement of Originality
This document is written by Luuk Mocking who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Abstract
The European Union (EU) has imposed sanctions on Russia as a reaction on the Ukrainian crisis. Russia does not seem to consider complying the EU’s demands, while some European countries have questioned the sanctions regime as sanctions are costly to both the target and the
sanctioner. The withdrawal of sanctions without any kind of compensation seems counterintuitive. This research shows with the help of a game theoretic model that the implementation of sanctions later the withdrawal without any form of compensation can be rationally explained. While the sanctioner is, even after costly sanctions without compensation, better off compared to the status quo.
1. Introduction 2. Literature review
a. Economic Sanctions i. Costs of sanctions
ii. Effectiveness of sanctions b. EU-Russian Sanctions
i. EU-Russian trade
ii. EU economic sanctions and their effects iii. Russian economic sanctions and their effects 3. Models and discussion
a. Model as explained by Lacy & Niou (2004) i. The model
ii. Results & Discussion
b. The model expanded with an extra step i. The model
ii. Results & Discussion
iii. Implementation of the model 4. Conclusion
5. Reference list 6. Apendices
1. Introduction
In response to the annexation of Crimea and the destabilisation of Ukraine the European Union (EU), amongst other countries, imposed sanctions against Russia. The sanctions were a series of diplomatic measures and restrictive measures against various people associated with the Russian regime and against exchanges and economic
cooperation with Russia (Newsroom, 2016). Although some European countries have been more reluctant about the sanctions than others, on November 18th 2016the United States of America and the EU agreed to keep these sanctions as the situation in Ukraine had not improved (Croft, 2015; Wagstyl & Chazan, 2016).
Sanctions have been a part of defence economics for centuries. Kaempfer & Lowenberg (2007, p. 869) write that sanctions date back at least to the Megarian decree of Athens in 435 B.C. and were used by leaders like Napoleon and Thomas Jefferson to influence an opposing government without using military power and therefor avoiding domestic casualties. Hufbauer et al. (2007) have made a taxonomy of sanctions, where they describe a rapid increase in the amount of sanctions in the last decades. There were almost as many sanctions after the Cold War as in the first 90 years of the 20th century.
Dirksen (2015, p. 2) writes that the sanctions decreased the amount of dollars on the currency market which forced the Russian Central Bank to intervene to keep the Ruble’s value. But as the bank was running low on reserves it decreased the amount of interventions and increased the interest rate to 6.5%. The Central Bank could not avoid the Ruble to lose half its value in 6 months. The lowering oil price has also had an impact on the Russian economy. The Brent oil price has plummeted from 110$/barrel around 2014 to below 40$/barrel at the start of 2016 (Nasdaq, 2016). Dirksen (2015, p. 2) explains how the impact of the sanctions is not easily distinguished from the effect of the decreasing oil price, but the sanctions have contributed to the exchange rate
decrease.
Kaempfer & Lowenberg (2007, p.869) describe how sanctions alter the terms of trade of both the target and the coercer. The pain on the target’s economy is supposed to shift the target’s policy in favour of the sender’s demands, because the target is afraid to lose its citizens support for his policy. But the lack of improvement in the Ukrainian crisis increases the dissension within the EU about the use of said sanctions
EU’s economy. And sanctions have been proven to have a perverse effect of decreasing democracy and increasing nationalism, even increasing the political resistance to the senders’ demands(Drezner, 2011; Kaempfer & Lowenberg, 2007, p. 870). Furthermore, empirical studies have shown that sanctions are only successful in reaching their goal 5-34% of the time according to the definition of successful (Kaempfer & Lowenberg, 2007, p. 871). But according to Eaton & Engers (1999), these studies struggle with a problem of selection bias. First of all, cases where sanctions were threatened and not actually imposed are not part of the research, while sanctions can be effective even if they are not actually imposed. Leading to a negative selection bias. And secondly, sanctions are only imposed if the sender thinks there is a high chance of succeeding. Leading to a positive selection bias. Eaton & Engers (1999) call the observed sanctions in empirical studies “a very unrepresentative tip of an iceberg” (p. 410). I will further discuss this issue in the second part of my thesis.
Another approach on sanctions is a game-theory approach. Where a state is not seen as an interest group of their voters, but as a single-rational actor. This approach shows how the decisions of one country affect the other country’s decisions (Kaempfer & Lowenberg, 2007, p. 890). Several models have been made to analyse the process of sanctions (Eaton & Engers, 1999; Drezner, 2003; Lacy & Niou, 2004). These models analyse the process of sanctions and the possible acceptance of the sanctioner’s policy by the target. But none of the models allow sanctioners to withdraw their sanctions, while several member states of the EU hint at a possible withdrawal of the sanctions without Russian concessions. Imposing sanctions without anything in return seems counterintuitive, as sanctions will lead to welfare loss. This research will focus on the possibility of withdrawal, and research if the choice for withdrawal without compliance can be rationally explained. This will be done by adjusting the model of Lacy and Niou (2004) to allow for withdrawal of sanctions by the sanctioning nations and calculating the expected returns to see if withdrawal is a pure strategy Nash.
The first part of my literature review is written about sanctions in general, the second part will be about EU-Russian sanctions. After the literature review I will explain and discuss the model of Lacy an Niou (2004) and expand the model with the possibility of withdrawal, followed by my conclusion.
2. Literature review
2.a. Economic sanctions
Economic sanctions have been used for centuries to influence the target’s policy without using military force. Mack and Khan (2000, p. 281) describe how the main assumption of sanctions is that the economic damage caused by sanctions on citizens moves them to pressure the target government to change its policy in a way that matches the senders’ demands. This assumption has been questioned in recent economic literature (Drezner, 2011; Hufbauer et al., 2007; Kaempfer & Lowenberg, 2007; Mack & Khan, 2000), which I will discuss in a later paragraph of this chapter. I will first describe the different kinds of sanctions. Then the costs of said sanctions. Thereafter the effectiveness of sanctions including a discussion of empirical research.
Traditionally, sanctions were split into two groups (Drezner, 2011; Kaempfer & Lowenberg, 2007): Trade sanctions and investment sanctions. Trade sanctions are a restriction on imports from or exports to a target country. And investment sanctions are a restriction on capital flow to the target country or even mandatory disinvestment in the target country. As said before in the introduction, sanction have become more popular since the end of the Cold War (Hufbauer et al., 2007). The amount of sanctions and especially its side effects were starting to become a political problem to the
sanctioning nations (Drezner, 2011). An example of ‘collateral damage’ by sanctions is given by Drezner (2011). He describes how sanctions cut Iraq’s GDP in half and were responsible for 100.000 up to 227.000 deaths among young children, while the sanctions still failed to yield concessions. Causing this many civilian deaths without yielding success was a cause of political problems in the sanctioning nations. To overcome these political problems a third kind of sanction, the targeted sanction, has become more popular (Kaempfer & Lowenberg, 2007; Drezner, 2011). The targeted sanctions, so called ‘smart’ sanctions, target the elite without imposing excessive humanitarian costs or threatening the lucrative trade with the target country. 2.a.i. Costs of economic sanctions
The terms of trade for both the sanctioner as the target will worsen in case of trade restrictions. Kaempfer & Lowenberg (2007) make a model to show the impact on the terms of trade and the welfare effects of trade sanctions. They find that multiple
factors have an impact on the terms of trade change caused by sanctions. At first the size of a country, as large countries are more self-sufficient, they suffer less extensively from trade bans. Second is the share of the countries that sanction the target. If every country in the world sanctions Russia it causes more harm than when it is just one country, because they can still import from other countries. And last the extend of the trade between the target and sanctioner pre-sanctions. Because the target will find it easier to discover alternative markets or alternative sources of supply for the sanctioned goods if the extend of trade was low (Kaempfer & Lowenberg, 2007, p. 872).
The costs of investment sanctions is also analysed by Kaempfer and Lowenberg (2007). They describe that an increase in disinvestment or sanctions that prevent companies from further investing in a target country increases capital outflow. This outflow decrease the country’s asset price, thus increasing the rate of return on said assets. The target country’s investors will be more likely to invest domestically, therefor lowering the capital outflow. If the amount of foreign disinvestment is higher than the domestic investment it will also decrease the demands for domestic currency, therefor have a lowering pressure on the currency’s value. Investment sanctions can also have a perverse effect, if mandatory disinvestment causes investors to sell assets for a price that is too low, thus enriching the certain target country’s investors that might have ties with the regime (Kaempfer and Lowenberg, 2007, p. 878).
The costs of smart sanctions are low to countries as a whole because of the way they are designed. They are designed to affect the elite without punishing the normal citizen, trying to get compliance with the least amount of collateral damage (Drezner, 2011).
2.a.ii. Effectiveness of sanctions
The traditional view is that the sanctions on the target will change the target country citizens’ view on the target’s policies to be changed in favour of the sanctioners’ (Mack & Khan, 2000, p. 281). In consensus with the traditional view, Eaton and Engers (1999) find that the target is more likely to comply when the costs of the sanctions are high compared to the costs of the policy change in favour of the sanctioner. Moreover, a patient sanctioner is more likely to yield success, just like a patient target will decrease the chance of successful compliance.
But according to Kaempfer and Lowenberg (2007, p. 870) there is uncertainty as to how these costs are supposed to translate in the target’s policy change. As mentioned before, empirical studies show that sanctions are only able to yield concession in 5-34% of the time according to the definition of success (Kaempfer & Lowenberg, 2007, p. 871). Scholars have multiple explanations for these results. One of the explanations is that sanctions often increase the amount of political integration in the target country. Mack and Kahn (2000, p. 282) describe how the sanctioners rather than the target regime get the blame for sanction-induced suffering. According to Drezner (2011), sanctions are more likely to be successful if the target is a democracy, but at the same time sanctions trigger drops in democracy of target countries. The non-democratic regime of Iraq, where a minimum of 100,000 up to 227,000 children died due to the sanctions while they still had no success is an example of how sanctions is used as an example of how sanctions do not work on non-democratic regimes.
The smart sanctions that are designed to reduce the collateral damage are supposed to target regimes without imposing harm to the normal citizen. While they do seem to reduce the political problems, evidence suggests that smart sanctions are no better than normal sanctions at generating policy change in the target country (Drezner, 2011).
But other scholars point out that one of the main problems with empirical evidence on sanctions is that the empirical research is done on a biased sample of the total population (Drezner, 2003; Eaton & Engers, 1999; Kaempfer & Lowenberg, 2007; Lacy & Niou, 2007). For example, it only measures cases where sanctions have been imposed. Drezner (2003) points out that actors have an incentive to reach an agreement before actually sanctioning, as both the sanctioner as target are worse off in case of sanctions. The threatening phase of sanctions is not included in the empirical research, which leads to a negative selection bias. Sanctions are only observed when the threat has already failed to change the target’s behaviour (Lacy & Niou, 2004). Sanctioning is done to show that you are resolute, to keep your reputation for further threats. A second way the sample is biased negatively is described by Drezner (2003): When sender and target are expected to have future conflicts they are less likely to make any concessions, as concessions in the present might weaken their credibility in future conflicts. Therefor sanctions are more often imposed when sender and target are hostile to each other, while getting concessions from hostile targets is less likely. Eaton and
Engers (1999, p. 410) also describe an example positive selection bias as countries will only implement sanctions if they think they are likely to be successful. As sanctioning without success does not only harm the target, but also the sanctioner. The empirical research on sanctions is done on a biased portion of the actual cases, what Eaton & Engers (1999, p. 410) call “a very unrepresentative tip of an iceberg”. This is why this research will further focus on the game theoretic models instead of empirical research. 2.b. Russian sanctions
2.b.i. EU-Russian trade
The cooperation with Russia has been a divisive issue in the European Union, even before the Ukrainian crisis (Leonard, 2007). The EU can be split into two different preference groups. One group sees Russia as a potential partner and tries to westernise and democratise the nation with cooperation. The other group sees and treats Russia as a threat. According to them, Russia’s expansionism and hostility should be stopped by supporting nearby anti-Russian regimes and excluding Russia from certain trades. The EU has failed to change Russia in the Putin era, but Russia did have an impact on the EU (Leonard, 2007). These groups can also be observed in the implementation of sanctions against Russia, as some countries are more reluctant than others to impose sanctions (Croft, 2015). Russia’s main influence in Europe is in the energy sector, as Europe depends on Russia for both gas, oil and solid fuel imports. Furthermore, Russia makes deals with individual member states to undermine the common strategy used by the EU as a whole. In 2013, 41.2% of the gas imports, 33.7% of the oil imports and 29% of the solid oil imports in the EU came from Russia (Eurostat, 2016a). The EU is trying to be less dependable on Russia in recent years, in 2014 the gas imports have decreased to 37.5% and the oil imports to 30.4%,. But even if the EU succeeds to be less dependent on traditional energy sources, Russian companies are investing in the European energy sector (Leonard, 2007).
In 2013, trade with Russia was responsible for 6.9% of the extra-EU exports and 12.3% of the extra-EU imports, which has decreased to 4.1% of the extra EU-exports and 7.9% of the extra-EU imports since the sanctions (Eurostat, 2016b). Although these percentages are not really high, it is a significant part of the European economy. Especially in certain sectors like the food export, where Russia in the second biggest
extra-EU market with a 10% market share (BBC, 2014b). The EU was Russia’s number one trading partner, accounting for almost 41% of all international trade in 2012 (BBC, 2014a). The EU exports transport equipment, chemicals and agricultural products. While the EU imports, like mentioned before, gas and oil. With a 41% portion of all trade in Russia, trade sanctions could have a huge impact on the Russia economy. But because the EU is very dependent on the oil and gas of Russia, it is unable to impose restrictive trade sanctions.
2.b.ii. EU economic sanctions and their effects
The EU’s economic sanctions on Russia can be divided among the three different groups of economic sanctions (Newsroom, 2016). At first the trade and investment sanctions.
Ban on imports of goods originating from Crimea, unless they have Ukrainian certificates.
Export ban on goods and services to Crimea on the transport, telecommunications and energy sector.
Ban on tourism services in Crimea.
Ban on import and export on military material to Russia.
Ban on investment in Crimea.
EU nationals and companies can no longer sell or buy new bonds, equity or financial instruments with a maturity of over 30 days issued by certain Russian banks and companies.
Ban on providing loans to certain Russian banks and companies.
A lot of trade and investment in Crimea is banned, as Crimea is considered to be unlawfully taken from Ukraine by Russia. The effects of the ban on trade and investment on Crimea are not easily measured, as Crimea used to be part of Ukraine and is now part of Russia. Furthermore the ban on military material to Russia, to prevent Russia from gaining military power to invade Ukraine.
The value of the Russian Ruble has known a steep decrease since the start of the EU’s sanctions (Dirksen, 2015), but the effect of the European investment sanctions against Russia is not easily distinguished from the effect of the lowering oil price. Dirksen
(2015) writes that the investment sanctions on Russia did have a lowering effect on the Ruble’s value. He explains that it worked as a leverage on top of the lowering oil price, combined leading to the currency crisis which Russia has experienced. There was no mandatory disinvestment, so the perverse effect of selling foreign assets for a too low price, favouring domestic investors was not present. This also restricted the costs of the EU’s sanctions to EU companies.
And last, a couple of smart sanctions were imposed to affect the Russian elite into changing their policy. As discussed before, this has only a small effect on the countries as a whole because of the way they are designed.
2.b.iii. Russian economic sanctions and their effects
Russia imposed sanctions on the EU as well, as a response to the European sanctions (BBC, 2014b). As a trade sanction Russia implemented a full embargo on food imports from the EU (among other western countries). A strong fall in exports of foods from the EU to Russia was due to the sanctions against the EU (Christen et al., 2015). The direct impact on the European economy was low due to previously signed contracts. But the long run potential trade loss is estimated at 92 billion Euro, with an additional negative effect on employment. As a result of this ban, Russia also had food shortages of certain products that were mainly imported from Europe (BBC, 2014c). Prices of fish had risen by 40% due to the food ban, impacting the Russian civilian. In this research, I will assume that the countersanctions by Russia will be lifted as soon as the EU’s sanctions are lifted.
As a smart sanction 89 European Politicians and military leaders are banned from visiting Russia (Guardian, 2015). The economic impact of the smart sanctions on the EU and Russia is low due to their design definition.
3. Model
3.a.i. The Model as explained by Lacy & Niou (2004)
The model of Lacy and Niou (2004) is used. There are 2 players, the coercer and the target. The coercer can decide to threaten sanctions, and later to impose sanctions. The target can decide to whether or not comply with the coercer’s demands (after the threat) and to whether or not to capitulate to the coercer’s demands (after the
imposement of the sanctions). For simplicity we assume that the compliance of the coercers demands by the target is binary, such that X = (x, ~x). We use x when the coercers demands are met, and ~x when there is no compliance. We also assume that the implementation of the sanctions is binary, S = (s, ~s), where s = sanctions and ~s is no sanctions.
The possible game outcomes are: O1: (no threat) : status quo
O2: (threat, compliance) : (x, ~s)
O3: (threat, no compliance, sanctions, no capitulation) : (~x, s) O4: (threat, no compliance, sanctions, capitulation) : (x, s) O5: (threat, no compliance, no sanctions) : (~x, ~s)
A target might have two different preference rankings. He can be resilient, and prefer sanctions over compliance or he can be compliant and prefer compliance over sanctions. They both prefer O5 to O1, because of the reputation loss for the coercer. The worst option for the compliant target is O3, where there are sanctions and no
compliance or capitulation. For the resilient target Lacy and Niou (2004) assume O4 is worse than O2, because the sanctions that were imposed had a negative impact. The resilient target always prefers not meeting the coercers’ demands over meeting the coercers’ demands, his preferences are separable and given by
(~x, ~s) > (~x, s) > (x, ~s) > (x, s), so ~x > x. The compliant target has non-separable demands given by (~x, ~s) > (x, ~s) > (x, s) > (~x, s).
The coercer also has two different preference possibilities. He can be resolute and prefer sanctions without compliance/capitulation over no sanctions or irresolute and prefer no sanctions over sanctions without compliance/capitulation. The resolute coercer has non-separable demands, given by (x, ~s) > (x, s) > (~x, s) > (~x, ~s). The irresolute coercer has separable preferences, (x, ~s) > (x, s) > (~x, ~s) > (~x, s), so ~s > s regardless the target complies on x. The irresolute coercer always prefers no sanctions over sanctions. But the target might comply after sanctions, so sanctions are still an option.
The players’ preferences from most favourable to least favourable are given below.
Irresolute coercer: O2 > O4 > O1 > O5 > O3 Resilient target: O5 > O1 > O3 > O2 > O4 Compliant target: O5 > O1 > O2 > O4 > O3
The model is visualized in figure 1. And the payoffs are shown in table 1.
Figure 1: visualization of model in extensive form Lacy & Niou (2004) p. 29
Both sides are allowed to be uncertain about the preferences of the other player. The odds of the coercer to be resolute are p and the odds of the target to be resilient are q. By reasoning backwards on the game-tree, if the coercer is resolute he should always sanction if the target does not comply. And the irresolute coercer should never impose sanctions. The resilient coercer never complies or capitulates and the compliant target always capitulates if he hasn’t complied yet. The remaining strategies are given below, for the coercer the resolute strategy is given first and the irresolute second. For the target, the first is the resilient strategy and the second is the compliant.
Coercer:
1 (Threaten/Sanction, Threaten/Sanction) = (TS, TS)
2 (Threaten/Sanction, Threaten/Do Not Sanction) = (TS, T~S)
Target:
A (Do Not Comply/Do Not Capitulate, Comply/Capitulate) = (~C~C, CC)
B (Do Not Comply/Do Not Capitulate, Do Not Comply/Capitulate) = (~C~C,~CC) 3.a.ii. Results & Discussion
The expected payoffs are now calculated and shown in table 2. The Nash
equilibriums are shown bold. The pure strategy Nash equilibriums are valid when the inequalities in table 3 are met. First, Nash strategy (TS, T~S; ~C~C, CC) is examined. For this Nash equilibrium to take place, the following inequalities have to be met.
(1) 𝑞𝑧4+ (1 − 𝑞)𝑧1 ≥ 𝑧3
(2) 𝑝𝑥4+ (1 − 𝑝)𝑥1 ≤ 𝑥3
Inequality (1) shows that the irresolute sanctioner is more likely to threaten if: (a) he expects the target to be compliant (as q is low), (b) the difference between
compliance and the status quo is big (𝑧1− 𝑧3 is high) or (c) the difference between backing down compared to the status quo is small (𝑧3− 𝑧4 is small). Inequality 2 shows
that the compliant target is more likely to comply if: (a) he expects the sanctioner to be resolute (p is high), (b) the difference between compliance and threat with no sanctions is relatively low (𝑥1− 𝑥3 is small) or (c) if the difference between compliance and
capitulation is relatively big (𝑥3− 𝑥4 is high).
The second Nash equilibrium (TS, ~T~S; ~C~C, CC) requires only the reverse of inequality (1) to be satisfied. This means that the irresolute sanctioner is less likely to threaten if: (a) the target is expected to be resilient (q is high), (b) the difference between not sanctioning after threat compared to accepting the status quo is big (𝑧3− 𝑧4 is high) or (c) the difference between compliance and the status quo is low
(𝑧1− 𝑧3 is small).
One of the main conclusions of Lacy and Niou (2004) is that the threat stage is critical for understanding and analysing sanctions. When sanctions are successful, success is usually made at the threat stage. When sanctions are imposed they often will not succeed in pure strategy equilibrium, as capitulation only happens in mixed game equilibriums.
Below are the five possible outcomes of the game, and how this outcome is possibly reached according to the equilibriums.
Outcome 1: Status quo.
Status quo is reached when the sanctioner does not threaten. This is a Nash equilibrium when the sanctioner is irresolute while the inverse of inequality (1) is satisfied. So he either expects the target to be resilient, the costs of not sanctioning after a threat are high or the difference between compliance and status quo is low.
Outcome 2: Threat, compliance. (x, ~s)
Outcome 2 is reached when the sanctioner threats and the target complies. The sanction threats when it is resolute or when inequality (1) is met. The target complies when it is compliant while the inverse of inequality (1) is met or both inequality (1) and (2) are met.
Outcome 3: Threat, no compliance, sanctions, no capitulation. (~x, s)
The sanctioner should be resolute and the target should be resilient. They always have the same strategy so it does not matter if inequalities are met.
Outcome 4: Threat, no compliance, sanctions, capitulation). (x, s)
This is not a Nash equilibrium, this can only take place in a mixed strategy equilibrium. Which is when inequality (2) is not satisfied. So when (a) the compliant target expects the sanctioner to be irresolute (p is slow), (b) the cost of capitulation compared to compliance are relatively low (𝑥3 − 𝑥4 is small) or the difference between
not complying with no sanctions and capitulation is big (𝑥1− 𝑥4 is high). Outcome 5: Threat, no compliance, no sanctions. (~x, ~s)
This will happen when the sanctioner is irresolute while at the same time both inequality (1) and (2) are met.
The results show that if sanctions are threatened it does not mean they will actually be imposed, and if they are imposed they are likely not to be successful, just like empirical research has shown. As capitulation is only an equilibrium in mixed strategies. And just like former research, this study shows that the threat of sanctions can be
without compliance can be seen as success, as the resolute coercer prefers sanctions over the status quo. While researchers base the success rate on compliance.
3.b.i The model expanded with an extra step
I will expand the model with an extra option for the coercer. The coercer will have the possibility to withdraw the sanctions, even if the target chooses not to capitulate. As discussed in the literature review, both the sanctioner and target
experience harm by sanctions in case of trade sanctions or investment sanctions. So a sanctioner might decide punishing the target is no longer worth is. Withdrawal is possible at any time after imposing the sanctions. But the sanctioner might choose to wait for the target to capitulate. An extra preference group is added to both the coercers and the targets to match the new preferences, the explanation of the new preference groups is given below.
The withdrawal of the sanctions is assumed to be binary, with w when the sanctions are withdrawn and ~w when the sanctions are not withdrawn. The new possible outputs are given below.
O1: No threat (status quo)
O2: Threat, Compliance (x, ~s, ~w)
O3: Threat, No compliance, sanctions, No capitulation, No withdrawal (~x, s, ~w) O4: Threat, No compliance, sanctions, capitulation (x, s, ~w)
O5: Threat, No compliance, No sanctions (~x, ~s, ~w)
O6: Threat, No Compliance, Sanctions, No capitulation, Withdrawal (~x, s, w)
The resolute coercer will be split into a real resolute and moderately resolute preference ranking. The real resolute coercer will always prefer the sanctions if the target has not met its demands, so it will prefer not to withdraw the sanctions. The moderately resolute coercer will prefer to withdraw the sanctions, as in his opinion the sanctions have proven its point.
The compliant target will split into a real compliant group and the moderately compliant group. The real compliant prefers to comply over sanctions with withdrawal, as the sanctions are costly and it prefers not to have a dispute with the coercer. We assume that the real compliant target prefers no capitulation & withdrawal over
capitulation, as he will not have to meet the coercer’s demands. The moderately resolute compliant prefers sanctions with withdrawal over compliance, as he will not have to meet the coercers’ demands.
The given preference rankings are given below. Real resolute: O2 > O4 > O3 > O6 > O1 > O5 Moderately resolute: O2 > O4 > O6 > O3 > O1 > O5 Irresolute: O2 > O4 > O1 > O5 > O6 > O3 Resilient: O5 > O1 > O6 > O3 > O2 > O4 Moderately compliant: O5 > O1 > O6 > O2 > O4 > O3 Real compliant: O5 > O1 > O2 > O6 > O4 > O3 Target Sanctioner
Outcome Resilient Moderately compliant Real Compliant Real Resolute Moderately Resolute Irresolute
O1, (status quo) w2 x2 v2 y5 a5 z3
O2, (x, ~s, ~w) w5 x4 v3 y1 a1 z1
O3, (~x, s, ~w) w4 x6 v6 y3 a4 z6
O4, (x, s, ~w) w6 x5 v5 y2 a2 z2
O5, (~x, ~s, ~w) w1 x1 v1 y6 a6 z4
O6, (~x, s, w) w3 x3 v4 y4 a3 z5
Table 4: Outcomes with given preference ranking payoffs.
Walking backwards on the game-tree we can conclude that the moderately
resolute coercer will always choose TSW as its strategy and the real resolute coercer will always choose TS~W. The irresolute target will always withdraw, but is unsure about the other preferences.
The resilient target will still choose not to comply and capitulate, as they will always be worse off when they do. The real compliant target will always capitulate but is unsure about complying and the moderately compliant strategy is unsure about both. The possible strategies are given below, the coercer’s strategies are given with the real resolute strategy first, the moderately resolute strategy second and the irresolute strategy last. The target’s strategies are given with the resilient strategy first, the moderately compliant strategy second and the real compliant strategy last.
Coercer: 1: (TS~W, TSW, TSW) 2: (TS~W, TSW, T~SW) 3: (TS~W, TSW, ~T~SW) Target: A: (~C~C, CC, CC) B: (~C~C, ~CC, CC) C: (~C~C, CC, ~CC) D: (~C~C, ~CC, ~CC) E: (~C~C, ~C~C, CC) F: (~C~C, ~C~C, ~CC)
The expected outputs of these strategies combined are calculated and shown in table 5. The first value is the expected output for the coercer and the second value is the expected output of the target. The Nash equilibriums are calculated and bolded. The Nash equilibriums are valid with the values shown in table 6.
3.b.ii. Results & Discussion
The outcomes of the possible games are visualised in table 7. Given the preference rankings (table 4), table 8 is formed. The Nash equilibriums are made by comparing the different tactics and shown bold. The expected payoffs are shown in table 5. Certain inequalities (inequalities (3) to (11)) must be met for the Nash equilibriums to be valid.
Table 6 and 9 show what inequalities are required for the specific Nash equilibrium to be valid. Every inequality is analysed in table 10.
(~C~C, CC, CC) (~C~C, ~CC, CC) (~C~C, CC, ~CC) (~C~C, ~CC, ~CC) (~C~C, ~C~C, CC) (~C~C, ~C~C, ~CC) (TS~W, TSW, TSW) O3 O6 O6 O3 O6 O6 O3 O6 O6 O3 O6 O6 O3 O6 O6 O3 O6 O6 O2 O2 O2 O4 O4 O4 O2 O2 O2 O4 O4 O4 O3 O6 O6 O3 O6 O6 O2 O2 O2 O2 O2 O2 O4 O4 O4 O4 O4 O4 O2 O2 O2 O4 O4 O4 (TS~W, TSW, T~SW) O3 O6 O5 O3 O6 O5 O3 O6 O5 O3 O6 O5 O3 O6 O5 O3 O6 O5 O2 O2 O2 O4 O4 O5 O2 O2 O2 O4 O4 O5 O3 O6 O5 O3 O6 O5 O2 O2 O2 O2 O2 O2 O4 O4 O5 O4 O4 O5 O2 O2 O2 O4 O4 O5 (TS~W, TSW, ~T~SW) O3 O6 O1 O3 O6 O1 O3 O6 O1 O3 O6 O1 O3 O6 O1 O3 O6 O1 O2 O2 O1 O4 O4 O1 O2 O2 O1 O4 O4 O1 O3 O6 O1 O3 O6 O1 O2 O2 O1 O2 O2 O1 O4 O4 O1 O4 O4 O1 O2 O2 O1 O4 O4 O1
Table 7: Outcomes with every tactic, Nash equilibriums are bold. The first row of every outcome is for the resilient target, the second for the moderately compliant and the third for the real compliant. The first column is for the real resolute sanctioner, the second for the moderately resolute sanctioner and the third for the irresolute sanctioner.
(~C~C, CC, CC) (~C~C, ~CC, CC) (~C~C, CC, ~CC) (~C~C, ~CC, ~CC) (~C~C, ~C~C, CC) (~C~C, ~C~C, ~CC) (TS~W, TSW, TSW)
y3,w4 a3,w3 z5, w3 y3, w4 a3, w3 z5, w3 y3, w4 a3, w3 z5, w3 y3, w4 a3, w3 z5, w3 y3, w4 a3, w3 z5, w3 y3, w4 a3, w3 z5, w3 y1,x4 a1, x4 z1, x4 y2,x5 a2, x5 z2, x5 y1, x4 a1, x4 z1, x4 y2, x5 a2, x5 z2, x5 y3,x6 a3, x3 z5, x3 y3, x6 a3, x3 z5, x3 y1,v3 a1, v3 z1, v3 y1, v3 a1, v3 z1, v3 y2,v5 a2, v5 z2, v5 y2, v5 a2, v5 z2, v5 y1, v3 a1, v3 z1, v3 y2, v5 a2, v5 z2, v5 (TS~W, TSW, T~SW)
y3, w4 a3, w3 z4,w1 y3, w4 a3, w3 z4, w1 y3, w4 a3, w3 z4, w1 y3, w4 a3, w3 z4, w1 y3, w4 a3, w3 z4, w1 y3, w4 a3, w3 z4, w1 y1, x4 a1, x4 z1, x4 y2, x5 a2, x5 z4,x1 y1, x4 a1, x4 z1, x4 y2, x5 a2, x5 z4,x1 y3, x6 a3, x3 z4, x1 y3, x6 a3, x3 z4,x1 y1, v3 a1, v3 z1, v3 y1, v3 a1, v3 z1, v3 y2, v5 a2, v5 z4,v1 y2, v5 a2, v5 z4, v1 y1, v3 a1, v3 z1, v3 y2, v5 a2, v5 z4, v1 (TS~W, TSW, ~T~SW)
y3, w4 a3, w3 z3,w2 y3, w4 a3, w3 z3, w2 y3, w4 a3, w3 z3, w2 y3, w4 a3, w3 z3, w2 y3, w4 a3, w3 z3, w2 y3, w4 a3, w3 z3, w2 y1, x4 a1, x4 z3,x2 y2, x5 a2, x5 z3,x2 y1, x4 a1, x4 z3,x2 y2, x5 a2, x5 z3,x2 y3, x6 a3, x3 z3,x2 y3, x6 a3, x3 z3,x2 y1, v3 a1, v3 z3,v2 y1, v3 a1, v3 z3, v2 y2, v5 a2, v5 z3, v2 y2, v5 a2, v5 z3, v2 y1, v3 a1, v3 z3, v2 y2, v5 a2, v5 z3, v2
Table 10: Inequalities of the adjusted model with the analysis of a possible difference, ceteris paribus.
Inequality Analysis
3 𝑞𝑧4+ (1 − 𝑞)𝑧1≥ 𝑧3 The irresolute sanctioner is more likely to threaten if (a) expects the target to be compliant or moderately compliant (1-q is high), (b) the difference between status quo and compliance is high (𝑧1− 𝑧3 is big) or (c) the difference between status quo and backing down after a threat is low (𝑧3− 𝑧4 is small).
4 (𝑝 + 𝑜)𝑣5+ (1 − 𝑝 − 𝑜)𝑣1≤ 𝑣3 A real compliant target is more likely to comply if: (a) it does not expect the sanctioner to be irresolute (1-p-o is small), (b) when the difference between compliance and threat without sanctions is low (𝑣1− 𝑣3 is small) or (c) the difference between capitulation and compliance is big (𝑣3− 𝑣5 is high).
5 𝑝𝑥6+ 𝑜𝑥3+ (1 − 𝑝 − 𝑜)𝑥1≤ 𝑥4 The moderately compliant target is more likely to comply if: (a) the sanctioner is not expected to be irresolute (1-p-o is small), (b) the difference between compliance and threat without sanctions is low (𝑥1− 𝑥4 is small) or (c) the difference between sanctions without withdrawal and compliance is big (𝑥4− 𝑥6 is high).
6 𝑝𝑥5+ 𝑜𝑥5+ (1 − 𝑝 − 𝑜)𝑥1≤ 𝑥4 The moderately compliant target is more likely to comply if (a) the sanctioner is not expected to be irresolute (1-p-o is small), (b) the difference between compliance and capitulation is big (𝑥4− 𝑥5 is high) or (c) the difference between compliance and threat without sanctions is low (𝑥1− 𝑥4 is small).
7 (𝑝 + 𝑜)𝑥4 ≥ 𝑝𝑥6+ 𝑜𝑥3 The moderately compliant target is more likely to comply if: (a) the difference between compliance and sanctions without withdrawal is big (𝑥4− 𝑥6 is high) or (b) the difference between compliance and sanctions with withdrawal is low (𝑥3− 𝑥4 is small).
8 𝑟𝑧1+ (1 − 𝑟)𝑧4 ≥ 𝑧3 The irresolute sanctioner is more likely to threaten if: (a) the target is expected to be moderately compliant (r is high), (b) the difference between compliance and status quo is big (𝑧1− 𝑧3 is high) or (c) the difference between threat without sanctions and compliance is low (𝑧3− 𝑧4 is small).
9 𝑞𝑧4+ (1 − 𝑞 − 𝑟)𝑧4≥ 𝑞𝑧5+ (1 − 𝑞 − 𝑟)𝑧2 The irresolute sanctioner is less likely to sanction if: (a) the difference between threat without sanctions and sanctions with withdrawal small (𝑧4− 𝑧5 is low) or (b) the difference between threat without sanctions and capitulation is big (𝑧3− 𝑧4 is high).
10 (𝑞 + 𝑟)𝑧4+ (1 − 𝑞 − 𝑟)𝑧1 ≥ 𝑧3 The irresolute sanctioner is more likely to threaten if: (a) the target is expected to be real compliant (1-q-r is high), (b) the difference between compliance and status quo is big (𝑧1− 𝑧3 is high)or (c) the difference between status quo and backing down after a threat is low (𝑧3− 𝑧4 is low)
11 𝑝𝑥6+ 𝑜𝑥3≥ 𝑝𝑥5+ 𝑜𝑥5 The moderately compliant target is less likely to capitulate if: (a) the difference between capitulation and sanctions is low (𝑥5− 𝑥6 is small) or (b) the difference between capitulation and sanctions with withdrawal is big (𝑥3− 𝑥5 is high).
Table 9: Which inequalities must be met for a Nash equilibrium to be valid. – Is inverse, so -3 is the inverse of inequality 3.
Table 7 shows the possible outcomes for the adjusted game. Unlike the model of Lacy and Niou (2004), every outcome is possible as a pure strategy Nash equilibrium in the adjusted model. Below are the outcomes with an example of an inequality (or inequalities) that must be satisfied for the Nash equilibrium to be valid.
Outcome 1: No threat (status quo)
The status quo is reached in the model for the irresolute sanctioner with either the inverse of inequality (3) and inequality (7) or the inverse of both inequality (7) and inequality (10).
So for example when the irresolute sanctioner expects the target to be compliant or moderately compliant and the moderately compliant target has a small preference difference between compliance and sanctions with withdrawal.
O2: Threat, Compliance (x, ~s, ~w)
Table 7 shows that compliance can be reached with every Nash equilibrium tactic, with every possible sanctioner and target, apart from the resilient target. For example with the (TS~W, TSW, T~SW; ~C~C, CC, CC) tactic, for every real compliant and moderately compliant target. The conditions that apply are inequality (3), (4), (5) and (6). So for example when the irresolute sanctioner is more likely to sanction because he expects the target to be compliant, the real compliant target is more likely to comply because the difference between compliance and threat without sanctions is low, the moderately compliant target is more likely to comply because the sanctioner is not expected to be irresolute and the moderately compliant target is more likely to comply the difference between compliance and capitulation is big. The other possibilities to reach compliance in Nash equilibrium with their conditions and analysis can be found in table 6, 7 and 8.
(~C~C, CC, CC) (~C~C, ~CC, CC) (~C~C, CC, ~CC) (~C~C, ~CC, ~CC) (~C~C, ~C~C, CC) (~C~C, ~C~C, ~CC)
(TS~W, TSW, TSW) - - -
-(TS~W, TSW, T~SW) 3 & 4 & 5 & 6 - -4 & 5 & 6 & 8 & 9 - 4 & 5 & 10 & 11 -(TS~W, TSW, ~T~SW) -3 & 7 - - - -7 & -10
-O3: Threat, No compliance, sanctions, No capitulation, No withdrawal (~x, s, ~w) As the moderately resolute and irresolute sanctioner always withdraw, this outcome is only observed with a resolute sanctioner. The resilient target always prefers sanctions over complying, so every game with a resolute sanctioner and resilient target has this outcome. It is also possible for the moderately compliant target to reach this outcome with a resolute sanctioner, for example at the (TS~W, TSW, ~T~SW; ~C~C, ~C~C, CC) tactic. This applies when the inverse of both inequality (7) and (10) are satisfied. So for example when the difference between compliance and sanctions without withdrawal is small and the target is not believed to be real compliant.
O4: Threat, No compliance, sanctions, capitulation (x, s, ~w)
This outcome is only observed as a Nash equilibrium at a real compliant target and a real resolute or moderately resolute sanctioner at the (TS~W, TSW, T~SW; ~C~C, CC, ~CC) tactic. The inequalities for this Nash equilibrium to be satisfied are the inverse of inequality (4) and inequalities (5), (6), (8) and (9). To summarize the analysis of these inequalities. This happens when the irresolute sanctioner is not likely to sanction and the moderately compliant target is believed to comply, making it worth it for the real compliant target to hope for the sanctioner to back down after the threat.
O5: Threat, No compliance, No sanctions (~x, ~s, ~w)
This is a Nash equilibrium outcome for the irresolute sanctioner and resilient target when the irresolute sanctioner threatens but does not sanction. For example at the (TS~W, TSW, T~SW; ~C~C, ~C~C, CC) tactic. Which happens when equilibriums (3) to (6) are satisfied. To summarize the analysis of these equilibriums, this happens when the irresolute sanctioner expects the target to be compliant, and compliant targets are more likely to comply.
O6: Threat, No Compliance, Sanctions, No capitulation, Withdrawal (~x, s, w) Sanctions with withdrawal is always the outcome when the sanctioner is moderately resolute and the target is resilient. It can also be a Nash equilibrium when the target is moderately compliant, when the moderately compliant target’s strategy is (~C~C). This happens, for example, at the (TS~W, TSW, ~T~SW; ~C~C, ~C~C, CC) strategy. This strategy is a Nash equilibrium when the inverse of both inequality (7) and
(10) are satisfied. This happens when, for example, the moderately compliant target is less likely to comply because the preference difference between compliance and sanctions with withdrawal is high. And the irresolute sanction is less likely to threaten because the target is not expected to be real compliant.
There are also 8 options where there is no pure strategy Nash equilibrium. Figure 2 shows a flow chart which can be used to determine if there is a valid pure strategy Nash equilibrium. If there is no pure strategy Nash equilibrium, players can always make a better decision so they have no best pure strategy, which only makes it possible to predict strategies in mixed equilibriums.
Figure 2: Flow chart to check if there is a valid pure strategy Nash equilibrium. Check if certain inequalities or inverses of inequalities are satisfied and follow the flowchart to get see if there is a Nash equilibrium.
The additional option of withdrawal adds an additional outcome that is observed in empirical research, which was not possible before in the old model. We find some things that are equal to the model of Lacy an Niou: if sanctions are threatened they are not always imposed and if sanctions are imposed they may not be successful. Which is in line with the selection biased empirical research. We also find that the model can be (partly) successful if the target does not comply or capitulate, because both the
moderately resolute sanctions and the real resolute sanctioner prefer sanctions with or without withdrawal over the status quo.
But there are also a few differences compared to their model. The first difference is that the adjusted model has every outcome as a possible pure strategy Nash
equilibrium. Apart from the new outcome with withdrawal, we also have capitulation as a pure strategy Nash equilibrium. There are five inequalities that must be satisfied, the target has to be irresolute and the sanctioner has to be either real resolute or
moderately resolute. Although the list of condition that have to be satisfied for the equilibrium to be valid is extensive, it is a pure strategy equilibrium.
The second difference is that this model has way more inequalities compared to the old model. This increases the amount of pure strategy Nash equilbriums compared to the old model, but it also makes the model more complex. The chance of having no pure strategy outcome has become higher. So the explanatory value of the model has also gone down, but this is compensated with the additional option of withdrawal and the fact that capitulation is now a viable option in pure strategy.
3.b.iii Implementation of the model
Sanctions are still imposed by the EU and Russia, and there has been no form of compliance. So we are currently in the sanction phase of the model, which allows the sanctioner to withdraw, but also allows the target to capitulate. In pure strategy equilibrium, this means the sanctioner is either moderately resolute or real resolute. The target can be either of the preference groups according to the model, but theory suggests (with imposed countersanctions) that there is a very high expectancy of Russia being resilient. Which leaves the EU with the choice whether to withdraw or not, as a resilient target will never capitulate. The moderately resolute and real resolute sanctioner always have the same strategy in this model: Withdrawal will be the pure strategy equilibrium output for the moderately resolute sanctioner and no withdrawal for the resolute sanctioner. As a moderately resolute sanctioner thinks that a period of sanctions can be enough to punish the target for their bad policy because it is also harmful to their own economy. Compared to the perpetual sanctions the real resolute sanctioner prefers. These outcomes are both preferred over the status quo. So if the EU were moderately resolute it would be a rational choice to withdraw the sanctions.
The EU and Russia both experience harm from the imposed trade and investment sanctions. Russia has known a severe currency crisis, partly caused by the investment sanctions. The EU has, for example, experienced the trade and employment loss caused by the full embargo on food imports by Russia. Some European countries are starting to question the sanctions as they are not going to succeed with their goal, capitulation by Russia. Which suggests they are moderately resolute. Another part of the EU is still in favour of sanctions, which suggests they are real resolute. Which brings us to a problem of analysing this casus. The model treats the sanctioner as one functioning body, while in fact, the EU is divided in their strategy approach. This make it impossible to predict what the EU will do, as it is not possible to say if the EU will reach consensus over preferring sanctions without withdrawal over withdrawal. Thus basically being real resolute over moderately resolute. But the model does show that either if the EU is real or moderately resolute, the output is a pure strategy Nash equilibrium, and it is better off than before sanctioning. The output will be sanctions with withdrawal if the EU is moderately resolute and sanctions without withdrawal if the EU is resolute.
The EU could also choose to only partially withdraw the sanctions, as some sanctions hurt their own economy more than others. An example of sanctions that hurt the European economy only to a small extend are the smart sanctions. This could be a way still have sanctions for the resolute countries and take away the collateral damage for the domestic economy.
4. Conclusion
This research has analysed the sanctions episode on Russia by the European Union. The first part of this thesis was a literature review where general sanctions, their costs and effectiveness were explained. We found that the sanctioner and target
experience harm of sanctions and the effectiveness of sanctions is low, but sanctions are still imposed. This is explained in the second part of the thesis where a model created by Lacy and Niou (2004) is analysed, which explaines why sanctions that are imposed without a capitulating target can still be a rational choice. But as the sanctions on Russia continue, member states of the EU have suggested to loosen the sanctions. The
withdrawal of sanctions without any form of compensation seemed counterintuitive as sanctions lead to welfare loss for both the sanctioner and the target.
But this research has demonstrated that the choice of withdrawal can be rational. Some countries prefer to give sanctions as a punishment but don’t want the sanctions to be perpetual. This preference group wants to implement sanctions, even if the target is not likely to capitulate, but will withdraw them later if they think they’ve proven their point. This has been done by creating an adjusted model where the option of possible withdrawal of sanctions was added to the model of Lacy and Niou (2004). Where expected outputs have been calculated. The pure strategy Nash equilibria show that withdrawal is a rational choice for the moderately resolute sanctioner. Not only can withdrawal be rational, the output is considered better than the status quo.
Finally the limitations of this research will be discussed. The first limitation of the model is that it doesn’t take reputation loss into account. While theory suggests that coercers will sometimes implement sanctions just to proof that they’re not always bluffing. That way, an irresolute sanctioner sometimes randomly sanctions. The second limitation is that the model treats the EU as one country with one preference, while in fact it is a group of countries with different preferences. And even the preferences of these countries can be changed over the time of the sanctions episode.
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6. Appendices Appendix 1.
Calculations equilibriums & Nash equilibriums model Lacy & Niou (2004). First the output for the coercer then the output for the target
1A (TS, TS) x (~C~C, CC) 𝑝𝑞𝑦3+ 𝑝(1 − 𝑞)𝑦1+ (1 − 𝑝)𝑞𝑧5+ (1 − 𝑝)(1 − 𝑞)𝑧1 𝑝𝑞𝑤3+ 𝑝(1 − 𝑞)𝑥3 + (1 − 𝑝)𝑞𝑤3+ (1 − 𝑝)(1 − 𝑞)𝑥3 2A (TS, T~S) x (~C~C, CC) 𝑝𝑞𝑦3+ 𝑝(1 − 𝑞)𝑦1+ (1 − 𝑝)𝑞𝑧4+ (1 − 𝑝)(1 − 𝑞)𝑧1 𝑝𝑞𝑤3+ 𝑝(1 − 𝑞)𝑥3+ (1 − 𝑝)𝑞𝑤1+ (1 − 𝑝)(1 − 𝑞)𝑥3
2A > 1A as 𝑧4 > 𝑧5 for the coercer 3A (TS, ~T~S) x (~C~C, CC) 𝑝𝑞𝑦3+ 𝑝(1 − 𝑞)𝑦1+ (1 − 𝑝)𝑞𝑧3+ (1 − 𝑝)(1 − 𝑞)𝑧3 𝑝𝑞𝑤3+ 𝑝(1 − 𝑞)𝑥2 + (1 − 𝑝)𝑞𝑤2+ (1 − 𝑝)(1 − 𝑞)𝑥2 3a > 2a if (1 − p)q(𝑧4− 𝑧3) + (1 − 𝑝)(1 − 𝑞)(𝑧1− 𝑧3) < 0 q(𝑧4− 𝑧3) + (1 − 𝑞)(𝑧1− 𝑧3) < 0 q𝑧4− 𝑞𝑧1+ 𝑧1− 𝑧3 < 0 q𝑧4+ (1 − 𝑞)𝑧1 < 𝑧3
If there is a high expectation of the target to be resilient it is better to go for 2a instead of 3a 1B (TS, TS) x (~C~C,~CC) 𝑝𝑞𝑦3+ 𝑝(1 − 𝑞)𝑦2+ (1 − 𝑝)𝑞𝑧5+ (1 − 𝑝)(1 − 𝑞)𝑧2 𝑝𝑞𝑤3+ 𝑝(1 − 𝑞)𝑥4 + (1 − 𝑝)𝑞𝑤3+ (1 − 𝑝)(1 − 𝑞)𝑥4 1B < 1A as 𝑥3 > 𝑥4 2B (TS, T~S) x (~C~C,~CC) 𝑝𝑞𝑦3+ 𝑝(1 − 𝑞)𝑦2+ (1 − 𝑝)𝑞𝑧4+ (1 − 𝑝)(1 − 𝑞)𝑧4 𝑝𝑞𝑤3+ 𝑝(1 − 𝑞)𝑥4+ (1 − 𝑝)𝑞𝑤1+ (1 − 𝑝)(1 − 𝑞)𝑥1
2B<2A if (𝑝𝑞𝑤3+ 𝑝(1 − 𝑞)𝑥4+ (1 − 𝑝)𝑞𝑤1+ (1 − 𝑝)(1 − 𝑞)𝑥1) < (𝑝𝑞𝑤3+ 𝑝(1 − 𝑞)𝑥3+ (1 − 𝑝)𝑞𝑤1+ (1 − 𝑝)(1 − 𝑞)𝑥3) 𝑝(1 − 𝑞)(𝑥4) + (1 − 𝑝)(1 − 𝑞)(𝑥1) < 𝑝(1 − 𝑞)(𝑥3) + (1 − 𝑝)(1 − 𝑞)(𝑥3) 𝑝𝑥4 + (1 − p)𝑥1 < 𝑥3 3B (TS, ~T~S) x (~C~C,~CC) 𝑝𝑞𝑦3+ 𝑝(1 − 𝑞)𝑦2+ (1 − 𝑝)𝑞𝑧3+ (1 − 𝑝)(1 − 𝑞)𝑧3 𝑝𝑞𝑤3+ 𝑝(1 − 𝑞)𝑥4 + (1 − 𝑝)𝑞𝑤2+ (1 − 𝑝)(1 − 𝑞)𝑥2 3a vs 3b 𝑥2 > 𝑥4 so 3a > 3b
Appendix 2
Calculations equilibriums & pure strategy Nash equilibriums adjusted model First the output for the coercer then the output for the target
1A: (TS~W, TSW, TSW) x (~C~C, CC, CC) 𝑝𝑞(𝑦3) + 𝑝 (𝑦1) + 𝑝(1 − 𝑞 − )(𝑦1) + 𝑞( 3) + ( 1) + (1 − 𝑞 − )( 1) + (1 − 𝑝 − )𝑞(𝑧5) + (1 − 𝑝 − ) (𝑧1) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧1) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤 ) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( ) 2A: (TS~W, TSW, T~SW) x (~C~C, ~CC, CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦1) + 𝑝(1 − 𝑞 − )(𝑦1) + 𝑞( ) + ( 1) + (1 − 𝑞 − )( 1) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧1) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧1) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤1) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( ) 3A: (TS~W, TSW, ~T~SW) x (~C~C, CC, CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦1) + 𝑝(1 − 𝑞 − )(𝑦1) + 𝑞( ) + ( 1) + (1 − 𝑞 − )( 1) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧 ) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤 ) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( ) 1B: (TS~W, TSW, TSW) x (~C~C, ~CC, CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦 ) + 𝑝(1 − 𝑞 − )(𝑦1) + 𝑞( ) + ( ) + (1 − 𝑞 − )( 1) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧1) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤 ) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( ) 2B: (TS~W, TSW, T~SW) x (~C~C, ~CC, CC)
𝑝𝑞(𝑦 ) + 𝑝 (𝑦 ) + 𝑝(1 − 𝑞 − )(𝑦1) + 𝑞( ) + ( ) + (1 − 𝑞 − )( 1) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧1) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤1) + (1 − 𝑝 − ) (𝑥1) + (1 − 𝑝 − )(1 − 𝑞 − )( ) 3B: (TS~W, TSW, ~T~SW) x (~C~C, ~CC, CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦 ) + 𝑝(1 − 𝑞 − )(𝑦1) + 𝑞( ) + ( ) + (1 − 𝑞 − )( 1) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧 ) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤 ) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( ) 1C: (TS~W, TSW, TSW) x (~C~C, CC, ~CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦1) + 𝑝(1 − 𝑞 − )(𝑦 ) + 𝑞( ) + ( 1) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧1) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧 ) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤 ) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( ) 2C: (TS~W, TSW, T~SW) x (~C~C, CC, ~CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦1) + 𝑝(1 − 𝑞 − )(𝑦 ) + 𝑞( ) + ( 1) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧1) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧 ) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤1) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( 1) 3C: (TS~W, TSW, ~T~SW) x (~C~C, CC, ~CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦1) + 𝑝(1 − 𝑞 − )(𝑦 ) + 𝑞( ) + ( 1) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧 ) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤 ) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( ) 1D: (TS~W, TSW, TSW) x (~C~C, ~CC, ~CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦 ) + 𝑝(1 − 𝑞 − )(𝑦 ) + 𝑞( ) + ( ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧 ) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤 ) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( ) 2D: (TS~W, TSW, T~SW) x (~C~C, ~CC, ~CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦 ) + 𝑝(1 − 𝑞 − )(𝑦 ) + 𝑞( ) + ( ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧 ) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤1) + (1 − 𝑝 − ) (𝑥1) + (1 − 𝑝 − )(1 − 𝑞 − )( 1) 3D: (TS~W, TSW, ~T~SW) x (~C~C, ~CC, ~CC)
𝑝𝑞(𝑦 ) + 𝑝 (𝑦 ) + 𝑝(1 − 𝑞 − )(𝑦 ) + 𝑞( ) + ( ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧 ) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤 ) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( ) 1E: (TS~W, TSW, TSW) x (~C~C, ~C~C, CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦 ) + 𝑝(1 − 𝑞 − )(𝑦1) + 𝑞( ) + ( ) + (1 − 𝑞 − )( 1) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧1) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤 ) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( ) 2E: (TS~W, TSW, T~SW) x (~C~C, ~C~C, CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦 ) + 𝑝(1 − 𝑞 − )(𝑦1) + 𝑞( ) + ( ) + (1 − 𝑞 − )( 1) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧1) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤1) + (1 − 𝑝 − ) (𝑥1) + (1 − 𝑝 − )(1 − 𝑞 − )( ) 3E: (TS~W, TSW, ~T~SW) x (~C~C, ~C~C, CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦 ) + 𝑝(1 − 𝑞 − )(𝑦1) + 𝑞( ) + ( ) + (1 − 𝑞 − )( 1) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧 ) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤 ) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( ) 1F: (TS~W, TSW, TSW) x (~C~C, ~C~C, ~CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦 ) + 𝑝(1 − 𝑞 − )(𝑦 ) + 𝑞( ) + ( ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧 ) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤 ) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( ) 2F: (TS~W, TSW, T~SW) x (~C~C, ~C~C, ~CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦 ) + 𝑝(1 − 𝑞 − )(𝑦 ) + 𝑞( ) + ( ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧 ) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤1) + (1 − 𝑝 − ) (𝑥1) + (1 − 𝑝 − )(1 − 𝑞 − )( 1) 3F: (TS~W, TSW, ~T~SW) x (~C~C, ~C~C, ~CC) 𝑝𝑞(𝑦 ) + 𝑝 (𝑦 ) + 𝑝(1 − 𝑞 − )(𝑦 ) + 𝑞( ) + ( ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧 ) 𝑝𝑞(𝑤 ) + 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + 𝑞(𝑤 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )𝑞(𝑤 ) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( )
Worse outcomes are deleted from the table, the remaining Nash equilibria are calculated: 2A<3A if (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧1) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧1) < (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧 ) 𝑞(𝑧 ) + (𝑧1) + (1 − 𝑞 − )(𝑧1) < 𝑞(𝑧 ) + (𝑧 ) + (1 − 𝑞 − )(𝑧 ) 𝑞(𝑧 ) + (1 − 𝑞)(𝑧1) < (𝑧 ) 2E<3E if (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )𝑧1 < (1 − 𝑝 − )𝑞(𝑧 ) + (1 − 𝑝 − ) (𝑧 ) + (1 − 𝑝 − )(1 − 𝑞 − )(𝑧 ) (𝑞 + )(𝑧 ) + (1 − 𝑞 − )𝑧1 < (𝑧 ) 3A<3E if 𝑝 (𝑥 ) + (𝑥 ) < 𝑝 (𝑥 ) + (𝑥 ) (𝑝 + )(𝑥 ) < 𝑝(𝑥 ) + (𝑥 ) 2A<2B if (𝑥 ) < 𝑝 (𝑥 ) + (𝑥 ) + (1 − 𝑝 − ) (𝑥1) (𝑥 ) < 𝑝(𝑥 ) + (𝑥 ) + (1 − 𝑝 − )(𝑥1) 2A>2C if 𝑝(1 − 𝑞 − )( ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )(1 − 𝑞 − )( ) > 𝑝(1 − 𝑞 − )( ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − )(1 − 𝑞 − )( 1) ( ) > 𝑝( ) + ( ) + (1 − 𝑝 − )( 1) 2A>2D if (𝑥 ) + (1 − 𝑞 − )( ) > 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − ) (𝑥1) + (1 − 𝑝 − )(1 − 𝑞 − )( 1) ( ) > 𝑝( ) + ( ) + (1 − 𝑝 − )( 1) (𝑥 ) > 𝑝(𝑥 ) + (𝑥 ) + (1 − 𝑝 − )(𝑥1) 2A>2E if 𝑝 (𝑥 ) + (𝑥 ) + (1 − 𝑝 − ) (𝑥 ) > 𝑝 (𝑥 ) + (𝑥 ) + (1 − 𝑝 − ) (𝑥1) (𝑥 ) > 𝑝(𝑥 ) + (𝑥 ) + (1 − 𝑝 − )(𝑥1) 2C>2B if 𝑝(1 − 𝑞 − )( ) + (1 − 𝑞 − )( ) + (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( 1) > 𝑝 (𝑥 ) + (𝑥 ) + (1 − 𝑝 − ) (𝑥1) + (1 − 𝑞 − )( )
> 𝑝( ) + ( ) + (1 − 𝑝 − )( 1) (𝑥 ) < 𝑝(𝑥 ) + (𝑥 ) + (1 − 𝑝 − )(𝑥1) 2C>2D if 𝑝(1 − 𝑞 − )( ) + (1 − 𝑞 − )( ) + (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( 1) > 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − ) (𝑥1) + (1 − 𝑝 − )(1 − 𝑞 − )( 1) (𝑥 ) > 𝑝(𝑥 ) + (𝑥 ) + (1 − 𝑝 − )(𝑥1) 2C>2E if 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − ) (𝑥 ) + (1 − 𝑝 − )(1 − 𝑞 − )( 1) > 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − ) (𝑥1) + (1 − 𝑝 − )(1 − 𝑞 − )( ) (𝑥 ) + (𝑝 + )(1 − 𝑞 − ) + (1 − 𝑝 − )(1 − 𝑞 − )( 1) > 𝑝 (𝑥 ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − ) (𝑥1) < 𝑝( ) + ( ) + (1 − 𝑝 − )( 1) (𝑥 ) > 𝑝(𝑥 ) + (𝑥 ) + (1 − 𝑝 − )(𝑥1) 2E>2B 𝑝 (𝑥 ) + (𝑥 ) + (1 − 𝑝 − ) (𝑥1) + (1 − 𝑞 − )( ) > 𝑝 (𝑥 ) + (𝑥 ) + (1 − 𝑝 − ) (𝑥1) + (1 − 𝑞 − )( ) > 𝑝( ) + ( ) + (1 − 𝑝 − )( 1) (𝑥 ) < 𝑝(𝑥 ) + (𝑥 ) + (1 − 𝑝 − )(𝑥1) 2E>2D 𝑝 (𝑥 ) + (𝑥 ) + (1 − 𝑝 − ) (𝑥1) + (1 − 𝑞 − )( ) > 𝑝 (𝑥 ) + 𝑝(1 − 𝑞 − )( ) + (𝑥 ) + (1 − 𝑞 − )( ) + (1 − 𝑝 − ) (𝑥1) + (1 − 𝑝 − )(1 − 𝑞 − )( 1) 𝑝(𝑥 ) + (𝑥 ) > 𝑝(𝑥 ) + (𝑥 ) > 𝑝( ) + ( ) + (1 − 𝑝 − )( 1)
