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Summary A Guide to Game Theory
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This summary contains the following chapters: 1, 2, 3, 4, 6, 8, 9. It contains all the necessary examples to make sense of the underlying theory.
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1Chapter 1 Game Theory Toolbox
The Idea of Game Theory
Hutton (1996) describes game theory as ‘an intellectual framework for examining what various parties
to a decision should do given their possession of inadequate information and different objectives’.
- This describes what game theory can be used for.
Game theory: a technique used to analyse situations where for two or more individuals (or
institutions) the outcome of an action by one of them depends not only on the particular action taken
by that individual but also on the actions taken by the other (or others).
- The plans or strategies of the individuals concerned will be dependent on expectations about
what the others are doing; decision making is interdependently related (called strategic
interdependence).
o Such situations are commonly known as games of strategy, or games, while the
participants in such games are referred to as players.
Because players in a game are conscious that the outcomes of their actions are affected by and
affect others, they need to take into account the possible actions of these other individuals when they
themselves make decisions. However, when individuals have limited information about other
individuals’ planned actions (their strategies), they have to make conjectures about what they think
they will do.
Describing Strategic Games
In order to be able to apply game theory a first step is to define the boundaries of the strategic game
under consideration. Games are defined in terms of their rules.
- The rules of a game incorporate information about the players’ identity and their knowledge
of the game, their possible moves or actions and their pay-offs.
- The rules of a game describe in detail how one player’s behaviour impacts on other players’
pay-offs.
- Pay-offs may be measured in anything that might be relevant to the situation. However, it is
often useful to generalize by writing pay-offs in terms of units of satisfaction or utility.
o Utility is an abstract subjective concept and its use is widespread in economics.
- When a strategic situation is modelled as a game and the pay-offs are measured in terms of
units of utility (sometimes called utils) then these will need to be assigned to the pay-offs in a
way that makes sense from the player’s perspectives.
o What usually matters most is the ranking between different alternatives.
o Sometimes it is simpler not to assign numbers to pay-offs at all, instead, we can
assign letters or symbols.
Rational individuals are assumed to prefer more utility to less and therefore in a strategic game a
pay-off that represents more utility will be preferred to one that represents less.
- This will always be true about levels of satisfaction or pleasure, whereas it will not always be
the case when we are talking about quantities of material goods (e.g., chocolate; it is possible
to eat too much chocolate).
Players in a game are assumed to act rationally if they make plans or choose actions with the aim of
securing their highest possible pay-off. This implies that they are self-interested and pursue aims.
However, because of the interdependence that characterises strategic games, a player’s best plan of
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action for the game, their preferred strategy, will depend on what they think the other players are
likely to do.
The theoretical outcome of a game is expressed in terms of the strategy combinations that are most
likely to achieve the players’ goals given the information available to them.
- Game theorists focus on combinations of the players’ strategies that can be characterised as
equilibrium strategies.
- If the players choose their equilibrium strategies they are doing the best they can given the
other players’ choices.
- The equilibrium of a game describes the strategies that rational players are predicted to
choose when they interact.
Games are often characterised by the way or order in which the players move.
- Simultaneous-move (static) games are games in which players move at the same time or their
moves are hidden.
- Sequential-move (dynamic) games are games in which the players move in some kind of
predetermined order.
Simultaneous-Move or Static Games
In these kinds of games players make moves at the same time or, what amounts to the same thing,
their moves are unseen by the other players. In either case, the players need to formulate their
strategies on the basis of what they think the other players will do.
Games can be diametrically opposed; if one wins the other effectively loses. These are games of pure
conflict. Often the pay-offs of the players in games of pure conflict add to a constant sum. When they
do the game is a constant-sum game.
- If the constant sum is zero, then it is a zero-sum game.
Most games are not games of pure conflict. In such games, there will be mutually beneficial or
mutually harmful outcomes so that there are shared objectives. These are sometimes called mixed-
motive games.
Pub Managers’ Game
Below is a pay-off matrix of different strategies by different pubs. This is a mixed-motive game.
Important: often, the pay-offs of the player whose actions are designated by the rows are written first.
So, in the example below, the pay-offs of Queen’s Head are shown first. Often, the colours are
matched of the text in the rows with the corresponding values (here these are shown in grey, versus
King’s Head’s values in black).
Table 1: Pay-off matrix for the pub managers' game
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Scores are subjective representations and therefore players’ pay-offs cannot be directly compared.
Penalty-Taking
The outcomes can also be represented by other things than scores (e.g., satisfaction level). To assess
this, you can attach values to different satisfaction levels (e.g., between 1-10, or 1-100) depending on
the range of the scale needed to represent the scores.
The example below is a simplification, in which the assumptions are made that either the player
scores, or the goalkeeper saves. Similarly, the player can only kick to the left, to the centre, or to the
right, the same holds for the movements of the goalkeeper. This would lead to the following pay-off
matrix.
- The cells of the pay-offs always add to the constant sum 10. It is a zero-sum game in this
case).
- All constant or zero-sum games are games of pure conflict and their outcomes are sometimes
difficult to predict. However, games of pure conflict won’t always be constant-sum games
although they can usually be represented in this way.
Table 2: Taking a penalty 1
Mixed strategies: strategies that mix up player’s pure strategies. In the example above, you can throw
a dice to decide which side you will kick as a player (e.g., throwing 1 and 2 would lead to left; 3 and 4
would lead to centre; 5 and 6 would lead to right; all of these options now would have a chance of
1/3).
- Mixed strategies can be useful in games of pure conflict, where one player does not want the
other to be able to predict their move.
Sometimes it is more convenient to write the players’ pay-offs as letters. This can be useful to
generalize the results of one piece of analysis to other similar but not identical games. In the penalty
game we could generalize the pay-offs in this way by substituting the letter W for the number 10 on
the assumption that W is greater than zero (W>0). Although the resulting game looks different than
the example above, in all important respects it is the same since W>0 (as denoted below the table).
Table 3: taking a penalty 1 with non-numerical pay-offs
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Sequential-Move or Dynamic games
In sequential-move games players make moves in some sort of order. This means one player moves
first and the other player or players see the first player’s move and can respond to it.
- The order of moves is important and the analysis of this type of game has to take this into
account.
- Sequential games are usually analysed using game trees or extensive forms.
In the example below, Apex (‘A’) and Convex (‘C’), choose between launching an advertising
campaign or not. The pay-offs represent the firm’s profits in thousands of euros.
Important: it is a convention that the pay-offs are written in the same order as the players’ moves.
Figure 1: The extensive form or game tree of Apex and Convex' advertising game
Repetition
Games that are only played once by the same players are called one-shot, single-stage or unrepeated
games. Games that are played by the same players more than once are known as repeated, multi-stage
or n-stage games, where ‘n’ is greater than one.
Meta-strategies: the moves the players plan to make at each repetition or stage of the game (in
repeated games).
Cooperative and Non-Cooperative Games
Essentially a game is cooperative if the players are allowed to communicate and any agreements, they
make about how to play the game as defined by their strategy choices are enforceable. Because
agreements can be enforced the players have an incentive to agree on mutually beneficial outcomes.
This leads cooperative game theory to focus on strategies that are implemented in the players’ joint or
collective interests. This is not the case in non-cooperative game theory where it is assumed that
players act only in their own self-interest.
Most of the games we will look at in this book are non-cooperative.
N-Player Games
N is the number of players in a game.
- If a game has two players, it is a 2-player game.
- If a game has more than two players, it is an N-player game.
Information
In some games players will be very well informed about each other but this will not be true in all
games. The information structure of a game can be characterised in a number of ways:
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