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Game Theory and Brexit Outcome Analysis

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These are some brief notes on applying some of the principles of game theory to analysing the current impasse in Britain in order to determine the likelihood of a no-deal/hard/disorganised Brexit.

The application of game theory principles can yield insight into the likely outcomes of apparently complex situations.

The application of the modified Mexican Standoff game to the Brexit negotiation process shows that you need relatively high probabilities that individual parties compromise in order to be guaranteed of achieving an overall compromise (based on the assumed three or more party threshold).

This contraintuitive reality leads to apparently rational groups of people collectively making decisions or allowing outcomes to occur that are so undesirable. Positive action rather than passive reaction are needed. But that positive action is regarded as a weakness and a sacrifice of the party’s preferred outcome.

The individual party strategy of holding out and not compromising seems individually reasonable. The individual party believes there is a greater probability that some or all of the other parties will compromise first. But that is not how probabilities work. When this strategy is applied collectively it leads to poor collective outcomes.

These are some brief notes on applying some of the principles of game theory to analysing the current impasse in Britain in order to determine the likelihood of a no-deal/hard/disorganised Brexit.

The application of game theory principles can yield insight into the likely outcomes of apparently complex situations.

The application of the modified Mexican Standoff game to the Brexit negotiation process shows that you need relatively high probabilities that individual parties compromise in order to be guaranteed of achieving an overall compromise (based on the assumed three or more party threshold).

This contraintuitive reality leads to apparently rational groups of people collectively making decisions or allowing outcomes to occur that are so undesirable. Positive action rather than passive reaction are needed. But that positive action is regarded as a weakness and a sacrifice of the party’s preferred outcome.

The individual party strategy of holding out and not compromising seems individually reasonable. The individual party believes there is a greater probability that some or all of the other parties will compromise first. But that is not how probabilities work. When this strategy is applied collectively it leads to poor collective outcomes.

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Game Theory and Brexit Outcome Analysis

  1. 1. Game Theory and Brexit Outcome Analysis Notes of applying game theory principles to an analysis of Brexit decision-making Alan McSweeney January 2019 http://ie.linkedin.com/in/alanmcsweeney
  2. 2. Game Theory and Brexit Outcome Analysis Page 2 Contents Introduction............................................................................................................................................................... 3 Current Standoff ........................................................................................................................................................ 3 Game Theory ............................................................................................................................................................. 4 Prisoner’s Dilemma.................................................................................................................................................... 4 Mexican Standoff ....................................................................................................................................................... 6 Modifying the Mexican Standoff to Analyse Brexit Outcomes .................................................................................. 9 Game of Chicken .......................................................................................................................................................12 Summary ..................................................................................................................................................................13
  3. 3. Game Theory and Brexit Outcome Analysis Page 3 Introduction These are some brief notes on applying some of the principles of game theory to analysing the current impasse in Britain in order to determine the likelihood of a no-deal/hard/disorganised Brexit. For the purposes of this analysis, I have assumed there are seven groups participating in the decision-making process: 1. Members of the Conservative party who want a second referendum/who want to remain in the EU 2. Members of the Conservative party who want to leave the EU with a deal such as remaining in the Customs Union 3. Members of the Conservative party who want to leave the EU without a deal or who are willing to tolerate the absence of a deal 4. Members of the Labour party who want a second referendum/who want to remain in the EU 5. Members of the Labour party who want to force a general election 6. Members of the Labour party who want to leave the EU with a deal such as remaining in the Customs Union 7. I have assumed that the EU is a single party who want to the UK to remain in the EU or leave the EU with a deal such as remaining in the Customs Union, accepting free movement of goods and people and paying some form of contribution I have assumed that there is not a group of members of the Labour party of sufficient size who want to leave the EU without a deal. I have omitted other parties from this analysis  Irish Government – I have assumed that they are observers with little direct influence on any party or their views are subsumed into the actions of the EU  DUP – I have assumed that they are observers with little direct influence on any party Current Standoff There is currently a standoff where none of the parties listed above wants to compromise on its core position. Party and Core Position Possible Outcomes if Party Compromises Members of the Conservative party who want a second referendum/who want to remain in the EU/defer its departure  Leave the EU with the current deal  Leave the EU without a deal  Delay leaving the EU Members of the Conservative party who want to leave the EU with the current deal  Leave the EU without a deal  Delay leaving the EU  Second referendum with possibility of reversal of leave decision Members of the Conservative party who want to leave the EU without a deal or who are willing to tolerate the absence of a deal  Leave the EU with the current deal  Delay leaving the EU  Second referendum with possibility of reversal of leave decision Members of the Labour party who want to force a general election  Conservatives remain in power  Leave the EU with the current deal
  4. 4. Game Theory and Brexit Outcome Analysis Page 4  Delay leaving the EU  Second referendum with possibility of reversal of leave decision Members of the Labour party who want a second referendum/who want to remain in the EU/defer its departure  Leave the EU with the current deal  Delay leaving the EU  Second referendum with possibility of reversal of leave decision Members of the Labour party who want to leave the EU with a deal such as remaining in the Customs Union  Leave the EU without a deal  Delay leaving the EU  Second referendum with possibility of reversal of leave decision Members of the Labour party who want to leave the EU without a deal or who are willing to tolerate the absence of a deal  Leave the EU with the current deal  Delay leaving the EU  Second referendum with possibility of reversal of leave decision I have assumed that the EU is a single party who want to the UK to remain in the EU with a deal such as remaining in the Customs Union or to postpone its departure or reverse its decision to leave  The UK leaves the EU without a deal  The UK delays leaving the EU  The UK leaves the EU with a deal that includes no long-term or definitive agreement on the Irish border I have omitted the outcome of the UK revoking its decision to leave the EU. The resolution of the impasse requires some of these parties to change their stances and possibly accept an outcome other than their preferred one. But given this unwillingness to change, the apparently most likely outcome would appear to be that no party or an insufficient number of parties changes their views so the default result is that the UK leaves the without a deal. For many of the parties this is the worst outcome. But it could be the most likely. Game Theory Game theory analyses the outcomes of interactions between (rational) decision-makers. It assumes that the all the parties participating in the decision-making have a defined strategy to which they adhere strictly, consistently and rationally. Each decision-maker will select an option based on the knowledge that the outcome is dependent both on their selection and the combined selections of the other decision-makers. One important aspect of game theory is the concept of equilibrium. This occurs when each decision-maker has selected a strategy and no decision-maker can benefit by changing strategies while the other decision-makers do not change theirs. So all parties are in equilibrium. The games used in game theory tend to be simplistic and not representative of the variability and irrationality of the real world. But nonetheless the analysis can yield insight into peoples’ behaviour. All the games tend to involve some or all of the following elements:  Brinkmanship and assessment of the strategy of other parties  Preference of winning over tying with other parties and tying over losing  A degree of risk and reward where the reward for winning may be less than the loss for losing Prisoner’s Dilemma Probably the most common and the simplest game analysed in game theory is the prisoner’s dilemma. It generally takes the form:
  5. 5. Game Theory and Brexit Outcome Analysis Page 5 Two criminals involved in the same crime are arrested. Each prisoner is in kept confined without any means of communicating with the other. There is insufficient evidence to convict the pair on the main charge that attracts a longer sentence but there is sufficient evidence to convict both on a lesser charge that attracts a shorter sentence. Each prisoner is offered a bargain. Each prisoner can therefore accept the bargain and agree to betray the other by giving evidence that the other committed the main crime or to remain silent. The outcomes are: 1. If A and B each betray the other, they will each serve a medium-duration (say two years) prison term. 2. If A betrays B but B remains silent, A will be set free and B will be held uniquely responsible for the crime and serve a longer duration (say three years) prison term. 3. If B betrays A but A remains silent, B will be set free and A will be held uniquely responsible for the crime and serve a longer duration (say three years) prison term. 4. If A and B both remain silent, both of them will only serve a short duration (say one year) prison term. In summary, the outcomes of the Prisoner’s Dilemma are: A Remains Silent Betrays His Partner B Remains Silent  Each serves 1 year  Prisoner B serves 3 years  Prisoner A goes free Betrays His Partner  Prisoner A serves 3 years  Prisoner B goes free  Each serves 2 years The rules of the game are:  Both prisoners are assumed to understand fully the nature of the game and be able to consider their situation rationally and unemotionally.  A and B have only one chance to either betray their partner or remain silent.  Neither party has any loyalty and animosity to the other.  Neither party will have any opportunity for revenge on the other for their betrayal.  Neither party will have the opportunity to reward the other for remaining silent. The logic of the outcome of the game is:  A should therefore betray B because:  If B remains silent, A betraying B will lead to A going free.  If B betrays A, A betraying B will lead to A receiving a sentence of two years.  Similarly B should betray A. However, this reasoning could be extended further. Both A and B reach the conclusion that betrayal appears to be the better option. Each knows the other has reached the same conclusion. So each knows they are going to serve two years in prison. Each could further rationalise that the other will remain silent knowing that both have realised that this will result in each serving one year in prison. This additional second-guessing (and possible subsequent N-guessing) is not considered in many examples of the Prisoner’s Dilemma. Though the assumption of perfect rationality tends to eliminate N-guessing because the outcome from this tends to be circular: I can betray because the other party will not betray. But then the other party reaches the same conclusion. So he betrays. So I betray. So we both lose. So I do not betray. The other party reaches the same conclusion. So he does not betray. And so on. As I mentioned above, the games used in game theory analysis tend not to represent real-world situations.
  6. 6. Game Theory and Brexit Outcome Analysis Page 6 Mexican Standoff While the stances of the various parties involved in Brexit have been described as a form of Mexican Standoff, this is not really correct. The classic Mexican Standoff involves the following set of rules:  There are N parties arranged in a closed loop.  Each party has a loaded gun pointed at the party ahead of them in the loop.  Each party has only one shot.  If a party shoots, the party ahead of them dies.  Guns are assumed to be fired simultaneously and instantaneously so even if a party is shot (and killed), they can still shoot (and kill) the party ahead of them in the loop before they are shot.  Every party will shoot or not shoot with the same probability.  No party can avoid being shot by, for example, ducking or running. This is not realistic for several reasons:  Each party will have a different probability of shooting  The first round of shooting could lead to a cascade of further shooting The following summarises the outcomes of thousands of simulations of a four-way Mexican Standoff. In this first example, there was a probability of 25% that any party will fire. Figure 1 - Four-Way Mexican Standoff The following chart contains seven values: 1. 0 of Group Dead – this is the percentage of the runs where no one was shot. 2. 1 of Group Dead – this is the percentage of the runs where one party was shot. 3. 2 of Group Dead – this is the percentage of the runs where two parties were shot. 4. 3 of Group Dead – this is the percentage of the runs where three parties were shot. 5. 4 of Group Dead – this is the percentage of the runs where four parties were shot. 6. Total % Dead – this is the percentage of parties shot 7. % Who Fired Dead – this is the percentage of parties who shot who were also shot
  7. 7. Game Theory and Brexit Outcome Analysis Page 7 Figure 2 - Summary of Outcomes of Mexican Standoff with Four Parties and Probability of 25% That Any Party Shoots The following chart summarises the outcomes where the probability that any party shoots is 50% Figure 3 - Summary of Outcomes of Mexican Standoff with Four Parties and Probability of 50% That Any Party Shoots If the probability that any party will shoot is P, then the probability that any individual party who shoots will die is P2. If P is small, P2 is very small. As P approaches 100% (certainty), P2 starts getting very close to P. This probability remains the same irrespective of the number of parties involved in the standoff. The following chart shows the probability distribution for a seven-way Mexican Standoff with a probability that any party shoots of 30%:
  8. 8. Game Theory and Brexit Outcome Analysis Page 8 Figure 4 - Summary of Outcomes of Mexican Standoff with Seven Parties and Probability of 30% That Any Party Shoots On average 30% of parties are shot. The probability that a person who shoots is also shot is 30%2 or 9%. This information could be used by any party in the following way:  If I feel that there is a low probability that any other party will shoot, then I can shoot with a very low chance (but still non-zero) of being shot.  Other parties might apply the same reasoning and feel they can get away with shooting the party ahead of them.  Therefore the probability that any party will shoot might be high.  Therefore the chance that I will die is high.  So I will not shoot.  Other parties will reach the same conclusion.  So no one shoots. Circular reasoning should be applied in the same N-guessing approach referred to above. The current state of Brexit stalemate cannot really be compared to this traditional Mexican Standoff. It is interesting to note that should you ever find yourself in a traditional Mexican Standoff, and you feel that there is a low probability that anyone will shoot and you are good at simple mental arithmetic, you can calculate the probability that you will be shot. You can also shoot the person ahead of you in the loop with a very low chance of being shot. You can modify the Mexican Standoff to include a possibility of a reactive or cascade round of shooting after the first shots have been fired. If no shots were fired during the first round, no shots will be fired during the reactive round. Assume that only those that are alive and those that did not fire the first time fire in this reactive round with a probability that different from the first round probability. The following shows the outcomes for a seven-way Mexican Standoff where the probability that any party shoots in the first round is 20% (call this P1) and the probability that any of the remaining parties shoots reactively is 30% (call this P2).
  9. 9. Game Theory and Brexit Outcome Analysis Page 9 Figure 5 – Seven Way Mexican Standoff with Cascade Shooting The probability that any party dies at the end of both rounds of shooting is: P1 + (1- 2 x P1)*(P2/2) The probability that any party who shoots is dead at the end of both rounds of shooting is: P1 2 + ((1- 2 x P1)*P2)2 So if you find yourself in a Mexican Standoff where you feel there could be a reactive round of shooting, you should practice you mental arithmetic before considering what action to take. Modifying the Mexican Standoff to Analyse Brexit Outcomes As it currently stands, the current state of the seven parties listed above in the Brexit process cannot be represented by a simple seven-way Mexican Standoff. Figure 6 – Seven-Way Mexican Standoff
  10. 10. Game Theory and Brexit Outcome Analysis Page 10 Each party is not directing their attention to just the party ahead of them in the closed loop. In order for a compromise to be reached, several parties much change their current positions and accept one of the possible less- desirable compromises listed above. The Mexican Standoff model could be changed in the following ways:  Each party does not shoot at the next in the loop resulting in the death of that party (much as some of the Brexit parties might like this) but publically states their willingness to compromise to a position other than their preferred one  Each party has the same probability of this first round compromise – call this P1  There is a cascade round referred to above where parties that have not stated their willingness to compromise can do so having possibly seen some parties compromising in the first round  Each party has the same probability of compromise in this cascade round – call this P2  The probabilities of compromise in each round are different  If no parties compromise during the first round, then no parties reactively compromise during the second For example, there could be a 15% probability of first-round compromise but a 30% probability of compromise in the second round. The following chart shows the range of likely outcomes for these two probabilities. Figure 7 – Modified Mexican Standoff with 15% Probability of Change in the First Round and 35% Probability in the Cascade Round The green bars show the individual percentages of the number of parties that compromise. So exactly 3 parties compromise 15.5% of the time. The purple bars show the cumulative number of parties who compromise. In this example, around there is a 36.13% probability that three or more parties compromise. The following shows the outcomes of the two rounds of compromises.
  11. 11. Game Theory and Brexit Outcome Analysis Page 11 Round 1 Outcome Round 2 Outcome Figure 8 – Sample Round 1 and Round 2 Outcomes Assume that to achieve compromise in the Brexit example, you need at least three of the seven previously listed parties to compromise. The following chart shows the probabilities of three or more parties compromising for various values of P1 (first round probability of compromise) and P2 (second round probability of compromise). Figure 9 - Probability That Three of More Parties Compromise For Various First and Second Round Compromise Probabilities The values in this chart are shown in the table below. Probability Of First Round Compromise 10% 20% 30% 40% 50% 60% 70% 80% 90% Probability Of First Round Compromise 10% 10% 28% 49% 68% 83% 93% 98% 100% 100% 20% 18% 39% 59% 76% 89% 95% 99% 100% 100% 30% 25% 47% 68% 82% 91% 97% 99% 100% 100% 40% 29% 54% 74% 86% 93% 98% 99% 100% 100% 50% 33% 59% 77% 88% 95% 98% 100% 100% 100% 60% 34% 61% 78% 90% 96% 98% 100% 100% 100% 70% 35% 62% 80% 90% 96% 99% 100% 100% 100% 80% 36% 62% 80% 91% 97% 99% 100% 100% 100% 90% 36% 63% 80% 91% 97% 99% 100% 100% 100%
  12. 12. Game Theory and Brexit Outcome Analysis Page 12 For example, if the probability that any party initially compromises is 20% and in response to this the probability that the remaining parties that did not compromise initially do so subsequently with an increased probability of 40%, the chances of getting the required threshold of three compromising parties is just 54%. Probability Of First Round Compromise 10% 20% 30% 40% 50% 60% 70% 80% 90% Probability Of First Round Compromise 10% 10% 28% 49% 68% 83% 93% 98% 100% 100% 20% 18% 39% 59% 76% 89% 95% 99% 100% 100% 30% 25% 47% 68% 82% 91% 97% 99% 100% 100% 40% 29% 54% 74% 86% 93% 98% 99% 100% 100% 50% 33% 59% 77% 88% 95% 98% 100% 100% 100% 60% 34% 61% 78% 90% 96% 98% 100% 100% 100% 70% 35% 62% 80% 90% 96% 99% 100% 100% 100% 80% 36% 62% 80% 91% 97% 99% 100% 100% 100% 90% 36% 63% 80% 91% 97% 99% 100% 100% 100% If you want to guarantee that three or more parties compromise, you need to have, for example, of the order of 50% of parties compromising initially and 50% of the remaining parties compromising in reaction to the actions of the parties in the first round. It may seem contraintuitive that you need such relatively high probabilities in order to be guaranteed of achieving an overall compromise (based on the assumed three or more party threshold). But it is precisely this that leads to apparently rational groups of people collectively making decisions or allowing outcomes to occur that are so undesirable. Positive action rather than passive reaction are needed. But that positive action is regarded as a weakness and a sacrifice of the party’s preferred outcome. This outcome illustrates the characteristics listed above:  Brinkmanship and assessment of the strategy of other parties  Preference of winning over tying with other parties and tying over losing  A degree of risk and reward where the reward for winning may be less than the loss for losing The individual party strategy of holding out and not compromising seems individually reasonable. The individual party believes there is a greater probability that some or all of the other parties will compromise first. But that is not how probabilities work. When this strategy is applied collectively it leads to poor collective outcomes. As with many game theory analyses, this represents a simplistic view of the real world. For example, the second round probability will probably vary with the value of the first round probability: the greater the first, the greater the second will be. All the compromises that each party can make are currently grouped as a single compromise outcome. They could be separated and assigned separate probabilities. There will also be multiple rounds. But the analysis and the outcomes it shows are instructive. Game of Chicken An alternative approach to modelling the strategies adopted by each of the Brexit parties is a modified form of the Game of Chicken. Here two parties are driving towards each other. The first to swerve loses. In summary, the outcomes of the Game of Chicken are: A Turn Remain on Course B Turn  A and B Tie  A Wins B Loses Remain on Course  A Wins B Loses  A and B Crash
  13. 13. Game Theory and Brexit Outcome Analysis Page 13 The Game of Chicken illustrates extreme brinkmanship. The cost of crashing is very high relative to the loss suffered if a party turns away. But if one party believes the other party to be reasonable, they may believe that party will turn and so not turn themselves. The current state of the Brexit could be compared to an N-way Game of Chicken. Figure 10 – N-Way Game of Chicken As with the Mexican Standoff, this approach would have to be modified to allow for two rounds of decision-making and an outcome of other than a crash. The results of an analysis of the outcomes of this game are similar to those generated by the modified Mexican Standoff. Summary The application of game theory principles can yield insight into the likely outcomes of apparently complex situations. The application of the modified Mexican Standoff game to the Brexit negotiation process shows that you need relatively high probabilities that individual parties compromise in order to be guaranteed of achieving an overall compromise (based on the assumed three or more party threshold). This contraintuitive reality leads to apparently rational groups of people collectively making decisions or allowing outcomes to occur that are so undesirable. Positive action rather than passive reaction are needed. But that positive action is regarded as a weakness and a sacrifice of the party’s preferred outcome. The individual party strategy of holding out and not compromising seems individually reasonable. The individual party believes there is a greater probability that some or all of the other parties will compromise first. But that is not how probabilities work. When this strategy is applied collectively it leads to poor collective outcomes. Based on this simple analysis, the current most likely outcome of a no-deal/hard/disorganised Brexit.
  14. 14. For more information, please contact: http://ie.linkedin.com/in/alanmcsweeney

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