Cruel Reality of Game Theory

Game theory is like a playbook for strategy. It’s a way to think about how people, businesses, and computer make their decisions when they’re competing with and against each other. It’s used a lot in fields like economics and computer science. In a modern context, it is used to study decision-making in all sorts of situations, such as playing chess against some of the best human players.

Sounds cool and posh, right? Think again after finishing reading this article.

As a branch of mathematics and study of science, game theory assumes all players in a game is playing rationally against other strategies. This means – everyone is playing against each other to achieve the best payoff, which is represented by gain minus loss. Because it is based on the assumption that opponents are playing rationally, so they will try to do the same as well.

For any arbitrary game that exist, if our interest is not in line with a specific party, then by theory, they should not act in our favour. In case of a zero-sum, two-player game, any gain for us means loss of equal value for our opponent. So under this context and basis, we are confident in introducing one of the fundamental algorithms – minimax (minimum-maximum).

Minimax is a way to choose the best current option by evaluating on behalf of all players respectively, which is choosing the best payoff for themselves. Its name come from its behaviour when dealing with zero-sum two-player game, where nodes representing us choose the best option for us, while those representing the opponent choose the worst option for us (equivalent to the best for them).

Below is an example of minimax algorithm in action, where square nodes represent maximising steps (our move), and round nodes represent minimising steps (opponent’s move). You can see how do nodes select its values on behalf of their respective positions, and how do they greedily fight for their interests.

We can observe from the tree above, when opponent is also playing rationally, we are bound to not choose the option containing the best possible payoff if we are not trying to worsen our returns. Providing this is the complete game tree that enumerates all the legitimate moves, outcomes, and payoffs, this tree have revealed a destiny of negative payoff – or in terms of game theory, losing the game. Because there is no chance, we are essentially trying to balance our needs and that of the opponent, in other words, fight for our profits while defending against the opponent.

Down to the results, minimax algorithm is seeking for an equilibrium, a set of actions that no parties would have no incentive to deviate from once the first action in this sequence is initiated. Again, because all parties are playing rationally, having no incentive means not continue playing along will lead to a decreased payoff, or losing the game altogether.

This tree has also shown another characteristic in games – one wrong move could cost highly, because rational opponent will leave no chance for greed. If we consider from the other side’s perspective at the leftmost child of the root (evaluated to be -3 for us), although the opponent is bound to win, but if they wrongfully chosen the other option for the protentional 13 payoff for them, they will loose at least 4 points, and most likely 10 points. We could also found out that the opponent will have no chance to recover from this if we are playing rationally.

In summary, under game theory, where players are playing rationally:

• Actions are chosen by balancing between offensive and defensive needs.
• There is no room for greed. Such actions may lead to unrecoverable worse outcome.

Above attributes are evident in a famous example – Prisoner’s Dilemma. Assume two baddie have collectively done something seriously bad, which would yield a punishment of 10 years for individual offenders, and 5 for participants within a group of offenders of a single act. The two baddies are now in a police station due to a minor offence, let’s say, ‘causing a public nuisance’. They do not explicitly know what do the police have on them, but the police think they are on something big, so they placed them in separate rooms and trying to get one or two them to confess. Both offenders know payoff for the them is distributed as following based on their choices:

Quite-evidently, because the cost of not testifying is low, the collective optimum for them would be to for both of them to keep silent and resist the lure of Tarekatsu-don.^[1] However, although they have agreed to not talk when this happens, they are unaware if the other party may be touched by the luxurious meal and come up with a story that their ex-comrade is the offender of all the nasty things they have done together and is willing to serve as a witness. So the only equilibrium here, unfortunately, is for both of them to testify against each other, despite the cost of staying silent is only a minor 0.1.

This is why I say the game theory is cruel, it is too rational and too cold that leads to probably some of the worst outcomes in certain situations. For example, in the example above, the two would become each others’ traitor and will probably be arch rival ever since. However, from the perspective of the greater good, we would actually love to see both of them getting convicted, as this is the only outcome that the justice has been served for both of them.

Another cruel part for game theory is, even in fields where it is believed to be applicable, sometimes it does not reflect how things work in real life.

Game theory has its calculations based on the assumption that all parties are playing according to the known rules – so in case there are unknown players involved, such as ‘chance’, or a hidden operator, the path it suggests could be invalid immediately. Furthermore, according to what we have discussed above, following those pathways could become very dangerous in such cases.

Examples of those outside influence are commonly seen in our daily lives – such as single-sided cheating, different application of rules depend on factors outside of the game, or employment of ungentlemanly conducts. This is not rare to witness – we have seen people using remote-controlled vibrating tool (anal beads) and succeeded, governments abusing their administrative powers to give local businesses unfair advantages, or for mafia to offer someone an ‘unrefusable offer’.

It would be undesirable to go through all the possibilities of such factors, or their impacts. In short, game theory represents methodologies of rationalism, but when it does not work, it uncovers realism and unfairness the society have. Unless this world would be as perfect as imagined, none of them will ever change.