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A game between an individual and a population: Introduction

已有 5274 次阅读 2012-2-6 03:27 |个人分类:thoughts|系统分类:科研笔记| 博弈, 群体, 细胞生物学

前几天似乎想明白了点什么,仔细一想又不太明白。一些已经想通了的,先记在这里。后面的再继续想。(亦公布于本人英文博客。)其实跟先前的狼-羊博弈同属多层博弈,但不同之处在于这里存在‘结构’。例子估计很好懂。我觉得将博弈的任一方扩充为一个群体,并且考虑这一方存在的内部博弈,可能对细胞、生化层次的分子间博弈有一定的参考价值。不过进一步的内容就敬请期待下回分解了~

2012年4月23日追记:昆明动物所王瑞武老师基于榕小蜂的一系列研究,已经包含了这个思路。参见这两篇论文Wang et al. 2009 (doi:10.1371/journal.pone.0007802), Wang et al. 2011 (doi: 10.1098/​rsif.2011.0063)。 看来我还得思考得再远一些,才能写出点东西。

Let me begin with a story read from the Metro newspaper. A fee was charged by the government on water use, but strongly resisted by the residents who were supposed to pay it. Many of the residents went to complain, to demonstrate and/or anything relevant showing their resistance, and finally made the fee canceled by the government. What a success!

However some of the residents who had been so obedient and had submitted the fee at the beginning, found themselves unable to claim their money back. One of the unfortunate, obedient people cried about this on the newspaper, but was quickly hit back by the other person who had fought for the cancellation. "You deserve that because you didn’t resist!"

The water fee story ends here, and I am not talking about the rightness of a resident’s choice or of the government’s policy. Instead, I am talking about the gaming strategies here adopted by different people, and more importantly, by the government.

Apparently, in a game like this, the residents have two candidate options, or strategies, i.e. to resist the fee, or to accept it and wait. However, in order to facilitate further discussion while avoiding some unnecessary misunderstanding in trivial things, I am modifying the scenario of the story here. All the residents have submitted the fee at the beginning, but then may choose to go on protest or to just wait, which are their two candidate strategies; meanwhile, the government will make a decision to pay the money back once some resident(s) go on protest, but also with two candidate strategies, i.e. to pay the exact amount only to those who have protested, or to everybody who have paid them. In the above case, the government chose the former strategy, and for residents obviously the optimal choice is to go to resist, because you will only lose if you give the money as required.

But what if the government shows some ‘conscience’ by paying all the money back to all the residents once the fee is cancelled? Let’s assume (and agree) there is a cost or risk when you go on demonstration or whatever to resist the fee, then the optimal choice will become otherwise. You would rather stay at home/work as regularly and safely, wait until somebody else successfully makes the fee cancelled by the government. You have the chance to get your money back, and loose nothing when compared with other residents!

Up to now, it is the gaming process between different residents. This is like a Boxed Pigs Game between multiple players, but there are no ‘big’ or ‘small’ pigs – all participants (residents) are equal in their cost and payoff with each strategy they may choose. Alternatively the case can also be seen as a multiple Prisoners' Dilemma, where the outcome is sometimes a typical Tragedy of the Commons. The government’s behavior is seen to have designed the rule of the game.

However, if we consider the government as the other player, then the whole thing becomes a two players' game, but one of the players is a POPULATION instead of a single person. In this new game, we have to consider not only the interaction between the two players, the residents and the government, but also the interaction among different individuals within one part of the players – the residents.

In this case, the government has two candidate strategies, i.e. to pay only the money back to those who have resisted (selfish), or to pay all the money back to everybody including those who had been obedient (show conscience), on the other hand each resident has two candidate strategies, i.e. to resist or to be obedient. We will see a straightforward outcome by letting both players ‘evolve’ naturally to find their ‘optimal’ or say ‘stable’ choices. The government will finally choose to ‘show conscience’, only because by doing this it has a higher opportunity to harvest an obedient population of residents where the money goes to the government's pocket. Correspondingly, the residents will all become obedient, because anybody who goes on protest will have a lower net payoff than other residents. 

As the result, although the government shows 'conscience', the residents will get nothing back because they don't resist. It's still a tragedy of the commons on behalf of the residents. At this state, the two players (the government and the residents) have achieved a Nash Equilibrium while all individual players within the resident population have achieved an Evolutionarily Stable Strategy (also a type  of Nash Equilibrium).

Here I have used only coarse-grained language to show this pattern of game playing, and the outcome above is not always valid depending on the values of various parameters, e.g. the cost/risk of resistance, the responding threshold of the government, the competition stress imposed at different levels, etc.

After all, such games with two players, one (or both) of which is actually a population where the intra-population game (actually competition in many cases) also happens, have provided a more complex game pattern. In such games, Nash Equilibriums have to be achieved at multiple levels to allow the whole system to stay at a steady state. This actually is a bridge between game theory and dynamical systems theory.

And what’s more, in the special case here, it is a structured game where a population of one player can only interact with an individual of the other player. This pattern is also seen in the game between mitochondrial and nuclear genomes, where many mitochondria can only interact with the nucleus in a single cell. 

I began thinking on the mito-nuclear issue around two years ago but only recently got some hint from the water fee case discussed on the newspaper in the last few weeks. I am yet dwelling on what molecular mechanisms may give rise to such a game, in terms of ATP production and allocation. More interesting points will be produced from this perspective, and advice is welcome from anyone.

This may be a new pattern of game where at least one of the players is actually a population. I checked it within some textbooks of game theory and didn’t find it ever mentioned. However I may be re-inventing the wheels. It is highly appreciated if anyone could let me know if you have seen the same thing proposed anywhere else.



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