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What Constitutes a Game?

There are a number of things that need to be in place to constitute a game:

In addition to these basic points, there is also another important factor in understanding game theory. Both players are aware of the other's possible moves. This is a situation, therefore, where in making a decision, each player will try to take account of what they think the other player will do.

Let us look at an example of what might be considered a 'game' and the elements listed above.

The above shows the well known game of tic-tac-toe, or noughts and crosses. The idea is to get a line of either Os or Xs vertically or diagonally to win the game. Let us assume we are playing the game - I go first and put my cross in the top left corner as shown. Where will you go?

If I am assuming that you are aiming to either win the game or, failing that, prevent me from winning, then the best place you can go is in the centre.

As a result of that move, it is highly likely, unless you or I make some horrendous mistake, that the game will end in a draw.

The principle of the game, however, is similar to that of game theory - there are two players, there is a strategy and we are both seeking to maximise a payoff - i.e. win the game or not lose.

Types of Games

Games can be of varying degrees of simplicity or complexity. At one end of the spectrum is the so-called 'zero-sum game'. This is a game where one player's gain is matched by the other player's loss. In such a game, it is impossible for both players to win - if one player receives a positive payoff, it must be matched by the other player's loss. The total benefits of 'winners' in the game is cancelled out by the sum of the losses made by others.

Chess - a classic two person zero-sum game - there can only be one winner. Copyright: Gaston Thauvin, from stock.xchng..

We can see examples of supposed zero-sum situations in many different economic scenarios. For example, we often hear complaints and lurid headlines about asylum seekers 'taking jobs' of British born workers. There is an assumption here that if an asylum seeker from (say) Afghanistan finds a job in the UK then that will mean a UK worker will lose their job - or not be able to gain it. We can also see this in pricing strategies between firms. If Tesco reduces its prices then there is an assumption that it will 'steal' customers from other stores such as Sainsbury.

Cooperative and Non-Cooperative Games

Can the process of finding a date be the subject of game theory? The movie, 'A Beautiful Mind' gave this impression although there is no evidence to suggest it had any basis in fact. Copyright: Olga Shevchenko, from stock.xchng.

Not all games, however, are zero-sum games. There can be many situations where there can be non zero-sum outcomes. In such a situation, the parties to an exchange can both be better off as a result of the trade. If we think about it there is a simple logic to this.

Let us say that I have a CD that I wish to exchange. If I want to get rid of it, we can assume that the satisfaction or utility that I now get from that CD has diminished. I need, however, to find someone who wants the CD I have but who also has a CD to exchange that I also want (this is called the 'double coincidence of wants'). You will be in exactly the same position as me with regard to the utility (satisfaction) you get from the CD you have to exchange. If we meet and agree to exchange CDs, we both leave the exchange better off - I have a CD I want and so do you - and we have replaced our existing CDs with ones that give us both greater utility.

Non zero-sum games provide a range of additional complexities. Remember that players come to the game with a strategy and a desire to maximise their payoff. It is possible that I could maximise my payoff by not cooperating with the other player/s in the game. Equally, it might be to my advantage to cooperate and in so doing maximise my payoff.

The movie "A Beautiful Mind", starring Russell Crowe, was the story of John Forbes Nash. You will hear more of Nash later in this article. Briefly, John Nash was a brilliant mathematician who turned his attention to game theory in the late 1940s and came up with a thesis that looked to solve the problem that had troubled economists and mathematicians for some time - how would rational bodies who were part of a potential bargain interact and split up the cake?

Nash identified the difference between cooperative and non-cooperative games. In the former, the players make decisions which are binding - in other words they will act on the decision they make. In the latter, each does not make such a commitment and any commitment to a strategy is not enforceable. Cooperative games can be seen in many national and international summits. Leaders meet and agree to a series of decisions and commit to carrying out these decisions on pains of being punished in some way through some agreed rule or law. If one party reneges on their part of the bargain, they face punitive punishment.

In the movie, one scene is used to highlight the basic idea behind cooperative games. Apologies, in advance, for the nature of the subject content! The example can be used swapping over the sexes as appropriate.

Two person Zero-Sum Games:

Let us look at an example of a two person zero-sum game.

The box below is referred to as a matrix. In this matrix, the numbers represent the payoff. Let us assume that the numbers represent the amount of money that each player might win or lose. The rows represent your three strategies. You can opt to go for row A, B or C. The three columns represent my three strategies - I can opt to go for column i, ii or iii.

The payoff is the box which forms the junction between the row you choose and the column I choose. So, if you pick row A and I choose column iii, I will pay you £2. If I selected column iii and you selected row A then you would pay me £2.

I have no idea what row you are going to choose and the same will apply to you with regard to my choice of column. However, there will be an assumption from both of us that we will choose the strategy that will maximise our payoff.

Row A provides you with the maximum payoff of £8 - but you will only get that if I choose column ii. I assume that you will choose row B. This is because whatever column I choose you will get a positive payoff. Choosing row A might get you £8 but could also lead to a 33% chance of you getting nothing or paying me £2. I would assume that you would rather have a guaranteed chance of getting at least £2, therefore B is the most rational decision for you to make.

Given that, where does this leave me? Choosing column i means I will have to pay you £7; choosing column ii involves me paying you £5. Therefore, my best option is to choose column iii which means I will have to pay you £2.

This does not represent a fantastic outcome for me but it is the best payoff I can hope for - I would rather give up £2 than either £7 or £5. In this case the game is zero-sum because the sum of the gains and losses by each player adds up to zero (you win £2 and I lose £2).

What we have been looking at here is what is known as the minimax theorem. This was devised by Van Neumann and states that every finite (in other words it has an end point somewhere) two-person zero-sum game has a value which represents the average payoff that one player can expect to 'win' from another if both played rationally.

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