A Guide to Game Theory
Chapter 1. Game theory toolbox
1.1 The idea of game theory
Game theory is a technique used to analyze 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). Thus
individuals in these kinds of situations are not making decisions in isolation, instead their
decision making is interdependently related. This is called strategic interdependence and
such situations are commonly known as games of strategy, or simply 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. These kinds of thought processes constitute
strategic thinking and when this kind of thinking is involved, game theory can help us to
understand what is going on and make predictions about likely outcomes.
Strategic thinking involves thinking about your interactions with others who are doing similar
thinking at the same time and about the same situation. Making plans in a strategic situation
requires thinking carefully before you act, taking into account what you think the people you
are interacting with are also thinking about and planning. Because this kind of thinking is
complex we need some sharp analytical tools in order to explain behavior and predict
outcomes in strategic situations – this is what game theory is for.
1.2 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 pay-offs. Players’ pay-offs
may be measured in terms of units of money or time, chocolate, beer or 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. Utility is an abstract, subjective concept and its use is
widespread in economics. Sometimes it is simpler not to assign numbers to pay-offs at all.
Instead we can assign letters or symbols to pay-offs and then stipulate their rankings. For
example, instead of assigning a pay-off of, say, ten to a bar of chocolate and three to a pizza,
we could simply assign the letter A to the chocolate and the letter B to the pizza and specify
that A is greater than B (i.e. A > B).
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 (i.e. they choose strategies to maximize
pay-offs). This implies that they are self-interested and pursue aims. However, because of
the interdependence that characterizes strategic games, a player’s best plan of 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 characterized as
equilibrium strategies. If the players choose their equilibrium strategies they are doing the
best they can give the other players’ choices. In these circumstances there is no incentive for
any player to change their plan of action.
,Games are often characterized by the way or order in which the players move. Games in
which players move at the same time or their moves are hidden are called simultaneous-
move or static games. Games in which the players move in some kind of predetermined
order are call sequential-move or dynamic games.
1.3 Simultaneous-move 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. We are
going to look at three examples: hide-and seek; a pub managers’ game; and a penalty-taking
game. The first of these is a hidden-move game and the second and third are simultaneous-
move games.
Both types of games are analyzed using the pay-off matrix or the strategic form of a game.
Often the pay-offs of the players in games of pure conflict (one wins, one loses) add to a
constant sum. When they do the game is a constant-sum game. Both hide and seek and the
penalty-taking game are constant-sum games. If the constant sum is zero the game is a
zero-sum game. Most games are not games of pure conflict. There is usually some scope for
mutual gain through coordination or assurance. In such games there will be mutually
beneficial or mutually harmful outcomes so that there are shared objectives. Games like this
are sometimes called mixed-motive games. The pub managers’ game is a mixed-motive
game.
1.3.1 Hide-and-seek
Hide-and-seek is played by two players called Robina and Tim. Robina chooses between
only two available strategies: either hiding in the house or hiding in the garden. Tim chooses
whether to look for her in the house or the garden. If Tim finds Robina in the time allotted he
wins €50, otherwise Robina wins the €50.
It shows all the possible pay-offs of the players that result from all their possible strategy
combinations. It is a convention that in each cell the pay-off of the player whose actions are
designated by the rows of the matrix are written first. The pay-offs of the player whose
actions are denoted in the columns are written second (Robina’s pay-off and strategies are
highlighted in blue).
1.3.2 Pub manager’s game
In the pub managers’ game the players are two managers of different village pubs, the King’s
Head and the Queen’s Head. Both managers are simultaneously considering introducing a
special offer to their customers by cutting the price of their premium beer. If neither pub
makes the discounted offer the revenue of the Queen’s Head is €7 000 in a week and the
revenue to the Kings Head is €8 000.
,The matrix shows that if the Queen’s Head manager makes the special offer his pay-off is 10
(i.e. €10 000) if the King’s Head manager also makes the offer, and 18 if he doesn’t. Similarly
if the King’s Head manager makes the offer his pay-off is 14 if the Queen’s Head manager
also makes the offer, and 20 if he doesn’t.
1.3.3 Penalty taking
If the striker scores he will be covered in glory and if the goalkeeper saves the penalty it will
be he who is covered in glory. This time the pay-offs cannot really be measured in terms of
money – being covered in glory is not really quantifiable in this way. Instead the pay-offs are
best represented in terms of levels of subjective satisfaction or utility. We can assume that if
the striker misses, his satisfaction level is zero and if he scores, the goalkeeper’s satisfaction
level is zero. I will use a pay-off of 10 to represent sky-high utility.
Notice that in the cells of Matrix 1.5 the pay-offs always add to the constant sum 10 since if
one player’s pay-off is 10 the other’s is zero. Games like penalty-taking and hide-and-seek
are called constant-sum games. If the constant sum in question is zero then the game is a
zero-sum game. All constant or zero-sum games are games of pure conflict (one loser, one
winner). However, games of pure conflict won’t always be constant-sum games although
they can usually be represented in this way.
1.4 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. This means that the order of moves is important and the analysis of this type of
game has to take this into account. It is not always easy to do this using pay-off matrices and
therefore
sequential games are usually analyzed using game trees.
In this version of that game the two firms, Apex and Convex, choose between launching an
advertising game or not. Apex moves first but the success of Apex’s campaign depends on
what Convex does. A, C1 and C2 represent the decision points in the game. Apex’s choices
are represented by the two branches that are drawn coming from the decision point or node
labelled A. As Apex moves first this point is the first decision point in the game, the first point
at which any player makes a move. At this point Apex chooses between launch or not
, launch. Whatever, Apex decides Convex sees Apex’s move and can respond. If Apex
launches its
campaign the game moves to C1 where Convex decides whether to launch its campaign or
not knowing full well that Apex has launched its campaign.
The payoffs are written on the far right of the diagram. It is a convention that the payoffs are
written in the same order as the players’ moves, i.e. the pay-off of the player who moves first,
in this case Apex, is written first.
1.5 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. The strategies of the
players in repeated games need to set out the moves they plan to make at each repetition or
stage of the game. These kinds of strategies are called meta-strategies.
1.6 Cooperative and non-cooperative games
Whether a game is cooperative or not is a technical point. 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. Most of the games we will
look at in this book are non-cooperative even though in some of them players choose
between cooperating with each other or not. But being able to choose to cooperate does not
make a game cooperative in the technical sense as such a choice is not necessarily binding.
1.8 Information
The information structure of a game can be characterized in a number of ways. The
categories used in this book are perfect information, incomplete information and asymmetric
information. If information is perfect then each player knows where they are in the game and
who they are playing. If information is incomplete then a pseudo-player called ‘nature’ or
‘chance’ moves in a random way that is not clearly observed by all or some of the players. If
not all the players observe the chance move then the information is also asymmetric. When
information is asymmetric not all players have the same information. Instead some player
has private information.
Chapter 2. Moving together
Introduction
Games in which the players choose their strategies and make their moves at the same time
or their moves are hidden from each other, are called simultaneous-move, static- or hidden-
move games. In games like this where the players’ moves are hidden from each other and in
games where the players move simultaneously a player’s choice cannot be made contingent
on another player’s actions. Players therefore need to reason through the game from their
Chapter 1. Game theory toolbox
1.1 The idea of game theory
Game theory is a technique used to analyze 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). Thus
individuals in these kinds of situations are not making decisions in isolation, instead their
decision making is interdependently related. This is called strategic interdependence and
such situations are commonly known as games of strategy, or simply 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. These kinds of thought processes constitute
strategic thinking and when this kind of thinking is involved, game theory can help us to
understand what is going on and make predictions about likely outcomes.
Strategic thinking involves thinking about your interactions with others who are doing similar
thinking at the same time and about the same situation. Making plans in a strategic situation
requires thinking carefully before you act, taking into account what you think the people you
are interacting with are also thinking about and planning. Because this kind of thinking is
complex we need some sharp analytical tools in order to explain behavior and predict
outcomes in strategic situations – this is what game theory is for.
1.2 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 pay-offs. Players’ pay-offs
may be measured in terms of units of money or time, chocolate, beer or 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. Utility is an abstract, subjective concept and its use is
widespread in economics. Sometimes it is simpler not to assign numbers to pay-offs at all.
Instead we can assign letters or symbols to pay-offs and then stipulate their rankings. For
example, instead of assigning a pay-off of, say, ten to a bar of chocolate and three to a pizza,
we could simply assign the letter A to the chocolate and the letter B to the pizza and specify
that A is greater than B (i.e. A > B).
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 (i.e. they choose strategies to maximize
pay-offs). This implies that they are self-interested and pursue aims. However, because of
the interdependence that characterizes strategic games, a player’s best plan of 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 characterized as
equilibrium strategies. If the players choose their equilibrium strategies they are doing the
best they can give the other players’ choices. In these circumstances there is no incentive for
any player to change their plan of action.
,Games are often characterized by the way or order in which the players move. Games in
which players move at the same time or their moves are hidden are called simultaneous-
move or static games. Games in which the players move in some kind of predetermined
order are call sequential-move or dynamic games.
1.3 Simultaneous-move 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. We are
going to look at three examples: hide-and seek; a pub managers’ game; and a penalty-taking
game. The first of these is a hidden-move game and the second and third are simultaneous-
move games.
Both types of games are analyzed using the pay-off matrix or the strategic form of a game.
Often the pay-offs of the players in games of pure conflict (one wins, one loses) add to a
constant sum. When they do the game is a constant-sum game. Both hide and seek and the
penalty-taking game are constant-sum games. If the constant sum is zero the game is a
zero-sum game. Most games are not games of pure conflict. There is usually some scope for
mutual gain through coordination or assurance. In such games there will be mutually
beneficial or mutually harmful outcomes so that there are shared objectives. Games like this
are sometimes called mixed-motive games. The pub managers’ game is a mixed-motive
game.
1.3.1 Hide-and-seek
Hide-and-seek is played by two players called Robina and Tim. Robina chooses between
only two available strategies: either hiding in the house or hiding in the garden. Tim chooses
whether to look for her in the house or the garden. If Tim finds Robina in the time allotted he
wins €50, otherwise Robina wins the €50.
It shows all the possible pay-offs of the players that result from all their possible strategy
combinations. It is a convention that in each cell the pay-off of the player whose actions are
designated by the rows of the matrix are written first. The pay-offs of the player whose
actions are denoted in the columns are written second (Robina’s pay-off and strategies are
highlighted in blue).
1.3.2 Pub manager’s game
In the pub managers’ game the players are two managers of different village pubs, the King’s
Head and the Queen’s Head. Both managers are simultaneously considering introducing a
special offer to their customers by cutting the price of their premium beer. If neither pub
makes the discounted offer the revenue of the Queen’s Head is €7 000 in a week and the
revenue to the Kings Head is €8 000.
,The matrix shows that if the Queen’s Head manager makes the special offer his pay-off is 10
(i.e. €10 000) if the King’s Head manager also makes the offer, and 18 if he doesn’t. Similarly
if the King’s Head manager makes the offer his pay-off is 14 if the Queen’s Head manager
also makes the offer, and 20 if he doesn’t.
1.3.3 Penalty taking
If the striker scores he will be covered in glory and if the goalkeeper saves the penalty it will
be he who is covered in glory. This time the pay-offs cannot really be measured in terms of
money – being covered in glory is not really quantifiable in this way. Instead the pay-offs are
best represented in terms of levels of subjective satisfaction or utility. We can assume that if
the striker misses, his satisfaction level is zero and if he scores, the goalkeeper’s satisfaction
level is zero. I will use a pay-off of 10 to represent sky-high utility.
Notice that in the cells of Matrix 1.5 the pay-offs always add to the constant sum 10 since if
one player’s pay-off is 10 the other’s is zero. Games like penalty-taking and hide-and-seek
are called constant-sum games. If the constant sum in question is zero then the game is a
zero-sum game. All constant or zero-sum games are games of pure conflict (one loser, one
winner). However, games of pure conflict won’t always be constant-sum games although
they can usually be represented in this way.
1.4 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. This means that the order of moves is important and the analysis of this type of
game has to take this into account. It is not always easy to do this using pay-off matrices and
therefore
sequential games are usually analyzed using game trees.
In this version of that game the two firms, Apex and Convex, choose between launching an
advertising game or not. Apex moves first but the success of Apex’s campaign depends on
what Convex does. A, C1 and C2 represent the decision points in the game. Apex’s choices
are represented by the two branches that are drawn coming from the decision point or node
labelled A. As Apex moves first this point is the first decision point in the game, the first point
at which any player makes a move. At this point Apex chooses between launch or not
, launch. Whatever, Apex decides Convex sees Apex’s move and can respond. If Apex
launches its
campaign the game moves to C1 where Convex decides whether to launch its campaign or
not knowing full well that Apex has launched its campaign.
The payoffs are written on the far right of the diagram. It is a convention that the payoffs are
written in the same order as the players’ moves, i.e. the pay-off of the player who moves first,
in this case Apex, is written first.
1.5 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. The strategies of the
players in repeated games need to set out the moves they plan to make at each repetition or
stage of the game. These kinds of strategies are called meta-strategies.
1.6 Cooperative and non-cooperative games
Whether a game is cooperative or not is a technical point. 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. Most of the games we will
look at in this book are non-cooperative even though in some of them players choose
between cooperating with each other or not. But being able to choose to cooperate does not
make a game cooperative in the technical sense as such a choice is not necessarily binding.
1.8 Information
The information structure of a game can be characterized in a number of ways. The
categories used in this book are perfect information, incomplete information and asymmetric
information. If information is perfect then each player knows where they are in the game and
who they are playing. If information is incomplete then a pseudo-player called ‘nature’ or
‘chance’ moves in a random way that is not clearly observed by all or some of the players. If
not all the players observe the chance move then the information is also asymmetric. When
information is asymmetric not all players have the same information. Instead some player
has private information.
Chapter 2. Moving together
Introduction
Games in which the players choose their strategies and make their moves at the same time
or their moves are hidden from each other, are called simultaneous-move, static- or hidden-
move games. In games like this where the players’ moves are hidden from each other and in
games where the players move simultaneously a player’s choice cannot be made contingent
on another player’s actions. Players therefore need to reason through the game from their
