The EU-UK Signaling Game

| Diquan Xian | May 9, 2019 |

Since 2017 March, the United Kingdom has engaged in official withdrawal from the European Union so that it can preserve its policy-making authority especially over the movements of people. Particularly, the withdrawal could be in two forms: hard Brexit or soft Brexit.

If the UK would insist on leaving both the single market and customs union within the EU and only maintain some degree of relations with Europe, it would move toward hard Brexit, which could cause significant damage to the UK’s relationship with the EU. Soft Brexit, by contrast, could alleviate Brexit-induced shocks to future EU-UK relations. If the UK would value its relationship with the EU sufficiently and consider staying in either or both of the single market and customs union, it would attain soft Brexit. Moreover, there would be normal free movements of people between the UK and the EU, and EU citizens in the UK, as well as UK citizens in the EU, could have preserved rights to access to public benefits. Hence, soft Brexit is similar to hard Brexit because of their common goal to leave the EU to ensure the UK’s political autonomy. They are different, nevertheless, because the former attempts to prevent disruptions in future EU-UK economic and social relations as much as possible through a series of agreements established before withdrawal, but the latter fails to do so and thus creates obstacles that prevent the EU and UK from participating in mutually beneficial activities as before in the future. Nonetheless, achieving soft Brexit requires more efforts to negotiate and gain ratification from parliament.

As one of the major negotiation topics, Northern Ireland border issues have posed a significant challenge to both sides of the argument. The Northern Ireland Protocol contains potential mechanisms to solve the issues, and it is specified in the draft withdrawal agreement endorsed by the current UK government and the EU last year. If it were confirmed, actual borders between Northern Ireland and Ireland were unnecessary because Northern Ireland would follow the EU’s trade rules when trading with Ireland, a member of the EU. Although the protocol could prevent hard borders, which is a commitment of both the EU and UK, it could not be a final solution because the UK did not accept it, but the EU has insisted on it. This has stimulated conflicts between them, which ultimately triggered the UK parliament’s rejection of the entire withdrawal agreement.

The conflict is that the UK would like additional negotiations with the EU to change the protocol in order to satisfy eurosceptic members of its parliament and members from Northern Ireland’s major party so that they would ratify a withdrawal agreement. The EU, nevertheless, refused new negotiations because they consider the protocol as an already well-negotiated solution to avoid hard borders. This situation has landed the UK in intricate positions where it struggles between making a relatively soft withdrawal to avoid hard Brexit and somewhat using hard Brexit to threaten the EU for better deals. Likewise, the EU has complicated circumstances because it would like to permit longer time for the UK to reach more agreements that could avoid hard Brexit, while also somewhat denying the possibility of a new extension so that the UK would have fewer options. Such interesting scenario motivates this article to use a signaling game model to understand the situation better because the author hypothesizes that both the EU and UK would prefer soft Brexit, but neither would like to reveal its determination to achieve it. In particular, the EU would act first and send signals to the UK that would then consider which action would be better: claiming hard Brexit or compromising for soft Brexit. Thus, a signaling game model is used to capture information hidden in those actions in order to learn how signals sent from the EU influence the UK’s strategies about stepping down to attain soft Brexit or insisting on threats of hard Brexit.

Figure1: Model for the EU-UK signaling game “N” in the center refers to nature, which includes unknown factors that predetermine how likely that EU prepares for hard Brexit. For the EU’s actions, “not approve” refers to “not approve new extension”, and “approves” refers to “approves new extension”. For the UK’s actions, “Hard” refers to “Hard Brexit”, and “Soft” refers to “Soft Brexit”. Variable c refers to the costs of the EU’s pretension to be prepared.


In the model illustrated above, the EU is the more informed player and the UK is the less informed player, because information about whether the EU is prepared for hard Brexit is fully grasped by the EU itself, but the UK may not capture all of it or not find it completely reliable. Regardless of whether the EU is prepared for hard Brexit or not, it could either approve a new extension or not. Specifically, a new extension in the model refers to potential future extensions after the current deadline of October 31, 2019. For the UK, no matter what type of the EU it is interacting with and whether it is permitted more time, it could either make soft Brexit happen or stick to hard Brexit.

This model also reflects the uncertainty of the EU-UK signaling game. Before the EU takes actions, the UK has initial beliefs about how likely that the EU prepared for hard Brexit, and is used to denote the probability of interacting with an unprepared EU. After observing the EU’s approval or rejection of a new extension, the UK could gain more information about the EU’s type, and adjust its beliefs and strategies accordingly. The model uses to denote the probability of the EU not approving a new extension and to indicate the probability of the UK insisting on hard Brexit. Moreover, numbers on the sides represent payoffs that both players receive after taking a certain combination of actions, where the first number is the EU’s payoff and the second number is the UK’s. This model hypothesizes that both the EU and UK would prefer a relatively soft withdrawal and thus makes payoffs of soft Brexit larger than those of hard Brexit for both players, except in the two cases of the upper part where the UK could benefit from using hard Brexit as a threat. The reason behind this hypothesis is that among several Brexit alternatives evaluated by UK parliament members, proposals that were close to winning support from a majority were relatively soft, such as advocating to stay within the customs union or join the European Free Trade Association so that the UK’s economic connections with the EU can be maintained. The EU would also prefer soft Brexit to lighten heavy repercussions on its economic activities in the UK. Therefore, if the EU was unprepared for hard Brexit but still rejected a new extension, it would need to bear a cost of. This is because given that no future extension is granted, the UK would have to leave at the designated time although with no or few agreements. Such hard withdrawal could reduce the EU’s payoffs, and the amount of reduction can be seen as the cost, but if the EU was prepared, the cost could be avoided because it would have taken necessary actions in private to protect itself against hard Brexit.

Solutions to this model are three types of equilibria. In the first one (separating equilibrium), given sufficiently high costs of rejecting new extension (c ≥ 8), if the EU was prepared, it would not approve a new extension, and if it was not, it would approve one. Meanwhile, regardless of its initial beliefs, the UK would learn that a prepared EU would not extend the deadline and that an unprepared one would. In this situation, a prepared EU could demonstrate its authority by fixing the deadline and expect that the UK would avoid hard Brexit as much as it could within the remaining time. Nonetheless, when the EU was unprepared and permitted a new extension, the UK could somehow exploit the EU by sticking to hard Brexit until EU would offer beneficial incentives for the UK to consider making its withdrawal softer. Therefore, it is essential for the UK to evaluate the EU’s level of preparation for hard Brexit so that it can weight which is better: the outright pursuit of soft Brexit or attain it with the help of hard Brexit.

In the second equilibrium (pooling equilibrium), even though it actually did not prepare for hard Brexit, the EU would pretend to be well-prepared, so that the EU would not approve a new extension no matter what. Such circumstance is possible given that costs of fixing the deadline are relatively low (c ≤ 8), and that the UK initially considered a prepared EU as the more likely type, as represented by (p ≤2/5). Hence, after observing the EU’s refusal to extend the deadline, the UK would realize that the EU might already take necessary actions to mitigate harmful consequences of hard Brexit and that it should prevent the situation where the UK alone is more vulnerable to hard Brexit from occurring.

In the third equilibrium (semi-separating equilibrium), the UK could not tell which of the two types of the EU it is interacting with, after observing how the EU dealt with the deadline. A prepared EU would find it beneficial to reject a new extension, but if it was not ready for hard Brexit, it would sometimes approve a new extension and sometimes not. The reason why an unprepared EU would play both strategies but with different probability is that costs of acting as if prepared are low (c<8), so it could impose pressure on the UK to figure out better withdrawal plans within the designated time. The EU’s pretension strategy could not sustain, nevertheless, because the UK initially thought it more likely to be unprepared (p>2/5), and therefore the EU in its best interests would authorize the UK’s extension request under some conditions. Specifically, the EU would reject the new extension with a probability of a* = (2-2p)/(3p), and this result shows that as the UK regarded an unprepared EU as the more likely type, the EU would prefer authorizing a future extension. Moreover, the UK could also randomize its two strategies in this circumstance because in addition to a single strategy of struggling for soft Brexit, it could take advantage of hard Brexit as leverage that would push the EU to compromise first so that some of its conditions could be satisfied. Such an opportunity arises when the EU did not prepare for hard Brexit, and UK would prefer threats of hard Brexit so that the EU would offer favorable conditions to make softer Brexit attractive to the UK. This occurs with a probability of b*= (10-c-5)/3, implying that as the EU’s pretension costs decline, the UK would be more likely to threaten to fight against the EU’s trick.

  p ≤2/5 p>2/5
(c<8) Pooling Equilibrium

l  EU (prepared): not approve new extension

l  EU (not prepared): not approve new extension

l  UK (sees “not approve”): Soft Brexit

l  UK (sees “approves”): Soft Brexit

Semi-Separating Equilibrium

l  EU (prepared): not approve new extension

l  EU (not prepared): not approve new extension with probability  a* = (2-2p)/(3p)

l  UK: Hard Brexit with probability  b*= (10-c-5)/3

(c≥ 8) Separating Equilibrium

l  EU (prepared): not approve new extension

l  EU (not prepared): approves new extension

l  UK (sees “approves”): Hard Brexit

l  UK (sees “not approve”): Soft Brexit

Separating Equilibrium

l  EU (prepared): not approve new extension

l  EU (not prepared): approves new extension

l  UK (sees “approves”): Hard Brexit

l  UK (sees “not approve”): Soft Brexit

Figure2: Results of the three types of equilibria

In general, the model focuses on the two major players the EU and the UK of the Brexit game and could capture important information about how the EU would shape the UK’s perceptions of itself by sending signals in order to influence the UK’s actions. Furthermore, the model could be improved to represent a broader scene by incorporating bargaining within the British parliament and external pressure such as from Irish domestic groups.

This March, Michel Barnier, European Chief Negotiator for Brexit affairs, suggested that “The U.K. should indicate a way forward before the 12th of April. Let me be frank: Without a positive choice, the default option would be a no-deal, which has become more likely. It was never our scenario, but the EU27 is now prepared.” Is the EU really prepared and will no longer authorize a new extension? Probably not, it still expects a positive response from the UK at least before the 31st of October.


Appendix: Calculations of the Three Types of Equilibria

Separating equilibrium:

For an unprepared EU to approve a new extension, its expected utility of approving one should be higher than that of not approving one, namely 2 ≥ 10-c, and (c≥ 8).

Pooling equilibrium:

If both a prepared EU and an unprepared one would not approve a new extension, the UK would consider itself interacting with a prepared type, after having a new extension rejected, because if the EU is prepared, it will always prefer to not approving a new extension in order to receive higher payoffs. In this case, the UK’s expected utility of soft Brexit should be higher than that of hard Brexit, namely 2p + 3(1-p) ≥ 5p + 1(1-p), and p≤ (2/5).

Semi-separating equilibrium:

In this case, an unprepared EU would sometimes approve a new extension and sometimes not, so the UK would need to know how likely that the EU was unprepared after it observed rejection of a new extension, which is the probability of an unprepared EU conditional on the observation that it did not approve a new extension, or P(not prepared | not approve = [P(not approve |not prepared)*P(not prepared)]/P(not approve) = ap/(ap+1-p) . Thus, the probability of a prepared EU conditional on the same observation is that P(prepared | not approve = 1-ap/(ap+1-p)=(1-p)/(ap+1-p). The purpose of the EU’s mixed strategies is to make the UK indifferent between hard Brexit and soft Brexit. This is represented by 5*ap/(ap+1-p) + 1*(1-p)/(ap+1-p)=2*ap/(ap+1-p) +3* (1-p)/(ap+1-p) and finally a*=(2-2p)/3p. Moreover, the UK also mixes its strategies to make the unprepared EU indifferent between approving a new extension and not, namely (4-c)b + (10-c)(1-b) = 2b + 5(1-b) and b* = (10-c-5)/3.

Diquan Xian is a junior majoring in economics with certificates in mathematics and computer science. As a summary of her two game theory classes, this article offers her a great opportunity to apply an important game theoretic model—signaling game model to analyze a real-world event. Above all, she is immensely grateful to her academic advisor Susan Hering, her article mentor Professor Susanne Mueller and Equilibrium’s managing editor Catherine Peterson, whose information about this writing platform, valuable suggestions about improving the model as well as the article, and advice on her writing language have made this article a qualified candidate for Equilibrium.