Games theory extended to multiple games

Introduction

The following question was put the the President of the World Bank, Dr. Jim Yong Kim by my MP, Geoffrey Clifton-Brown 

“How does the World Bank expect the global economy to take the painful steps necessary to address climate change when the world’s most powerful economic blocks remain locked in nuclear weapon standoffs which is terrifying them into borrowing trillions of dollars for weapons that will outlast civilisation as it collapses from climate change and which force preservation of business as usual to the very end?”

This question is considered in the context of Game Theory, where a game is a defined as a decision making scenario where players must take a decision such that the outcome to them is dependent on that made by the other players.

In the question above, the outcome to the players in a game (i.e. climate change) is also dependent on the outcome made in in other games (i.e. nuclear weapons). There are three fundamental games at play which are described below.

The first is the game of climate change negotiations which generally is assumed to be about obtaining an agreement to cut CO2 emissions. However, past failure to do so, means that this must now also be about agreeing to the adaptation measures to be taken. So far, despite all the hype of renewables, global consumption of fossil fuels is continuing to push atmospheric CO2 deeper into the danger zone. It is hypothesised that nations are compelled to do this to maintain economic and military advantage. This is backed up by past evidence at critical decision making points. For example, the US Congress voted unanimously against inclusion into the Kyoto 1 agreement because it would have constrained its military while not constraining its adversaries in the same way and in the Copenhagen COP China refused to make any commitments to cut back coal consumption as it was intent on out competing Western economies.

The second is the game that nuclear weapons states must play on the size and operational status of their arsenals; the decisions in this game have always been about how to make cuts in the size of the arsenals and to minimise the risk of a premature launch by taking weapons off high alert status while trusting others to do the same. These agreements must be made in a world becoming more unstable due to climate change and where the risk of a pre-emptive strike is increasing due nuclear weapons proliferation.

The third game, in reference to the borrowing needed to fund the pursuit of nuclear weapons in the above question, is about maintaining the debt based economic system that military industrialisation competition needs. The fundamental assumption behind its modus operandi is that there is no limit to growth and things that we cannot afford now can be paid in the future by virtue of continued economic growth. However, once this impossibility is to be acknowledged then an economic system such as a carbon rationing or a carbon taxation must be introduced. Without this no agreement on climate change will be reached as fossil fuel consumption will continue to rise and the tensions it causes will propel nations towards nuclear weapons. However, once this is imposed it makes funding a military industrial complex impossible, so security must come centre stage to the negotiations.

Thus the three games outlined above are connected in a deadly dilemma. In an attempt to understand the dynamics of interconnected games a series of experiments were run across thee maths classes which extended the concept of the prisoners dilemma.

The basis of the experiment was as follows:

A class was given the opportunity to win either £1.50 which they could share amongst themselves or one person could win a bar of chocolate, which has a monetary value of 70p.

The game consisted of splitting the class into competing pairs of students. Each student in each pair is given two cards, one says “I love you and want to work for you and will do anything for you,” the other says “XXXX you buddy.” See Appendix A for the cards.

The rules are simple:

If both students play “I love you and want to work for you and will do anything for you” the cost of their love is £2 each.

If both students play “XXXX you buddy” the cost of their love is £8 each.

If one student plays “I love you and want to work for you and will do anything for you” and the other plays “XXXX you buddy,” then the student who plays the “I love you” card gets charged £10 for his love as a punishment for being so stupidly trusting and the one that plays the “XXXX you buddy” card gets charged only £1 as a reward for his ruthless thuggery.

The objective is to minimise the cost of love and the dilemma is clear. If both players trust each other and play the “I love you card,” the total cost of their love is £4. If both mistrust each other and play the “XXX you buddy,” total cost of their love is £16 as they seek to minimise their individual costs.

To play the game, the combined cost of love over five rounds was calculated and if this was kept below a given level, then the class could share the prize of real money. If not the person with the lowest cost of love could get the chocolate bar.

Thus the challenge is that a player not only has to trust his competitor, but also has to trust the outcome from the games that other competing pairs are playing.

The payoff matrix replicates the dilemma of nations making decisions on climate change. Two nations could decide to pursue a zero-carbon economy and it might cost them say £2billion. However, if a nation’s competitor refuses to convert to zero carbon and pursues a fossil economy then the cost to the nation that opts for the zero-carbon economy rises to £10billion as a result of having to cope with the resulting ecological damage and the loss of competitive advantage, while the cost to the nation that maintains a fossil fuel economy is  only £1billion as it is able to seize food and resources from its weaker rival.  If on the other hand, both nations decide to maintain a fossil fuel economy, the minimum cost will be £8billion to both from the ecologic damage incurred, but by maintaining competitive advantage neither will be liable for the full cost.  The actual costs are immaterial, all that counts is the relative values with respect to the choice, see Appendix B for the pay-off matrix.

The results follow for three classes:

Class 1

The target was to get “the cumulative cost of their love” below £90 across four simultaneous games and over five rounds. If all students played the love card, the minimum cost of their love would be £80, thus allowing two players to default and still win the money.

The results follow:

Conclusion of the game

In the first round, one player in each game played the “XXXX you buddy” card. The result was that it would be impossible for the class to win the money. Players in Games 1 and 2 collaborated and agreed to stick to the pattern of one player playing “XXXX you buddy” and the other playing “I love you.” This ensured that one player would get the lowest possible score and so win the bar of chocolate; the cost for this is that the bar of chocolate would have to be shared with amongst all the players in Games 1 and 2.  

Players in games 3 and 4 were not party to this agreement and so got nothing at the end.

Once the result became a foregone conclusion and the target for the minimum cost of love could not be achieved, players effectively lost interest but carried on out of a sense of duty.

Implications

In this game, once the initial sub optimal positions were set across all the games it was difficult to move away from it. This reflects the difficulty that nations face in negotiations when they have to move significantly from the positions that they have previously taken based on self-interest to those that are in the best interests of all parties.

Thus globally, nations that have already committed to high carbon and militarised societies will become entrenched in these positions, not just because of the conversion difficulty, but also because of the responses from other players that will be determined on the results of past rounds.

Once the result becomes fixed, interest in the game diminishes. This was reflected in the last UK election where climate change was not considered, despite the scientific community screaming for urgent and extreme action. However negotiations continue out of a sense of duty, thus the UK will continue sending delegates to the climate change conferences despite the impossibility of achieving a satisfactory result.

Despite the groups being unable to co-operate across all the games, small scale co-operation was made between games 1 and 2 to share the suboptimal prize (the bar of chocolate). This is reflective of the co-operation that is seen between states who are close competitors. Thus, the European and US co-operated on trade pacts and military alliances while Russia and China likewise co-operate on military and energy policies. However in each case the win from the localised co-operation is far less than that obtainable from globalised co-operation. 

Class 2

The target was to get “the cumulative cost of their love” below £90 across four simultaneous games and over five rounds. If all students played the love card, the minimum cost of their love would be £80, thus allowing two players to default and still win the money.

The results follow:

Conclusion of the game

In the first three rounds all players co-operated to play the “I love you” card and were on track to keep the cost of their love below £90 and win the money prize.

However, in round 3 the co-operation fell apart. One player reneged on the agreement and by being the only player to play the “XXXX you buddy” card stole a lead on the rest of the players. In the last round, all players could still win the money, however the player who had previously played the “XXXX you buddy” was now incentivised to play the same strategy. If he played “XXXX you buddy” he would definitely win the bar of chocolate, even if someone else did the same.  This is exactly what he did. At the end of the game he graciously shared the bar with his opponent, who both ate it and left. The rest sat there bemused. 

This is the emergence of a free-for-all scenario. It occurs when one player reneges on an agreement that has only a minimal chance of delivering the optimum solution even if all the other players are still prepared to work towards the wider agreement.

Implications

China has already embarked on a free-for-all strategy. Its carbon emissions initially from coal, and now from oil, are massively out of proportion to the rest of the world. They have effectively played the “XXXX you buddy” card against the rest of the world. From the outside, it is as if they have already decided that there is no point in going for a climate change agreement, so they will race to get everything they can while they still can. It is a highly dangerous strategy. If everyone reciprocates, then no one will survive. Even if no nation follows it, no one will survive. It is of note that India is now following China’s path as its closest competitor.

Class 3

The target was to get “the cumulative cost of their love” below £65 across three simultaneous games and over five rounds. If all students played the love card, the minimum cost of their love would be £70, thus allowing two players to default and still win the money. In the last round the minimum cost of love was reduced to £65

The results follow:

Conclusion of the game

The dynamics of this group were considerably different to the others. The sat closer together and spent more time discussing strategies between them. Their success in the first round of getting all to agree along with the communication they had set up between them provided the basis for reinforcement such that it became difficult to change the pattern that had been established. It is a similar observation to that of the first class, except that class had become stuck on the sub-optimal solution.

As they entered the last round still with no defections, the target for the minimum cost of love was reduced to £65 to incentivise someone to defect. Even this did not break the pattern that had now emerged as the social pressure to comply was so much greater than the temptation to go for personal gain.

On winning the £1.50 the class immediately went to college shop and bought two bars of chocolate which they shared equally amongst each other.

The class acknowledged that they were only able to achieve this because they were working closely together and said that had each competing pair been sitting in different rooms they would not have been able to achieve this.  Their success may also have been enabled by only three simultaneous games being played, rather than four.

Implications

If multiple games are being played where the result from one can adversely affect the other, then the games must be interconnected to achieve the optimum result, hence climate change, nuclear weapons and economic reform talks must be fundamentally interconnected in the same way that success was achieved by Class 3 by working closely together across all three games.  

In these circumstances the optimum position can be achieved

By contrast the problems that Classes 1 and 2 identified was the tendency for the games to become easily stuck in a suboptimal position as soon as a single player in one game achieved competitive advantage in a single game and in so doing reduced the probability of a best collective agreement being achieved.

The challenge facing nations is that the prizes are somewhat different, and much more is at stake. Instead of the combined money prize of £1.50 for collaboration, the prize now is  survival. Instead of mendacious behaviour being rewarded by a chocolate bar that can be shared with nearest competitors, the prize for mendacious behaviour is that a nation will be able preserve wealth right up to the point of their inevitable extinction. Neither is a great result.

If any optimism can the taken from this, it is that a clear interconnection between the games being played does enable the best collective result to be obtained, but this must first overcome the entrenchment caused by past actions.

Appendix A - Playing cards

Appendix B - Pay off matrix