Summary A Guide to Game Theory
Overview of the book
Chapter 1: basic ideas and concepts underlying game theory
Chapter 2: simultaneous- or hidden-move games are analysed and the dominant strategy and Nash
equilibrium concepts are defined.
Chapter 3: prisoners’ dilemma
Chapter 4: how sequential decision making can be modelled using game theory and extensive forms
Chapters 6: Nash equilibrium concept is extended to incorporate randomising or mixed strategies
Chapter 8: more realism is incorporated by allowing for the possibility that people play some games
more than once
Chapter 9: the methodology used to analyse dynamic games in Chapter 4 is applied to strategic
bargaining problems
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Chapter 1 Game Theory Toolbox
1.1 The idea of game theory
Game theory is:
An intellectual framework
for examining what various parties to a decision should do
given their possession of inadequate information and different objectives
Game theory is 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). In these circumstances
the plans or strategies of the individuals concerned will be dependent on expectations about what
the others are doing. Thus individuals in these kinds of situations are not making decisions in
isolation, instead their decision making is interdependently related.
Strategic game: a scenario or situation where for two or more individuals their choice of action or
behaviour has an impact on the other (or others).
Strategic interdependence: individuals’ decisions, their choices about actions, impact on each
other and therefore their decision making is interdependently related.
Player: a participant in a strategic game.
Strategy: a player’s plan of action for the game.
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 knowledge of
the game, their possible moves or actions 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. 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. Payoffs can be
described as e.g. I like chocolate (A) more than pizza (B), but this is simpler A>B. 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. 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.
,Games are often characterised 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.
Pay-offs, equilibrium and rationality:
Pay-off: measures how well the player does in a possible outcome of a game. Pay-offs are
measured in either material rewards (money) or in utility that a player derives from a particular
outcome of a game.
Utility: a subjective measure of a player’s satisfaction, pleasure or the value they derive from a
particular outcome of a game.
Equilibrium strategy: a ‘best’ strategy for a player in that it gives the player his or her highest pay-
off given the strategy choices of all the players.
Equilibrium in a game: a combination of players’ strategies that are a best response to each
other.
Rational play: players choose strategies with the aim of maximising their pay-offs.
1.3 Simultaneous-move games
In these games players make moves at the same time or, what amounts to the same thing, their
moves are unseen by the other player. There are 3 types of games:
1. Hide and seek: e.g. a hide and seek game in a house. If Tim finds Robina in the time allotted he
wins €50, otherwise Robina wins the €50. The pay-off matrix is shown below. It shows all the
possible pay-offs of the players that result from all their possible strategy combinations. In each
cell the pay-off of the player whose actions are designated by the rows of the matrix are written
first (Robina). The pay-offs of the player whose actions are denoted in the columns are written
second (Tim).
2. Pub manager’s game: both managers are simultaneously considering introducing a special offer
to their customers by cutting the price of their premium beer.
3. Penalty taking: in this game 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. Notice that in the cells of Matrix the pay-offs
always add to the constant sum 10 since if one player’s pay-off is 10 the other’s is zero.
Conclusions of the games above:
Diametrically opposed: the first and third games as if one wins the other effectively loses. Games
like this are games of pure conflict.
, Constant-sum games: games in which the sum of the players’ pay-offs is a constant. If the
constant sum is zero the game is a zero-sum game. Constant-sum games are games of pure
conflict; one player’s gain is the other’s loss. E.g. Hide-and-seek and the penalty-taking game.
Mixed-motive games: A mix of pure strategies determined by a randomisation procedure. 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
1.4 Sequential-move or dynamic games
In sequential-move games players make moves in some sort of order. One of the players moves first
and another sees the first player’s move before deciding how to respond. It is not always easy to do
this using pay-off matrices and therefore sequential games are usually analysed using game trees.
The payoffs are written in the same order as the players’ moves, i.e. the pay-off of the player who
moves first (A) 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.
1.6 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. In a non-
cooperative games agreements cant being enforced.
1.7 N-player games
N is the number of players in the game. If a game has two players then it is a 2- player game. But if
there are more than two players then the game is an N-player game where N is greater than 2.
1.8 Information
The equilibrium strategies of the players will depend on what kind of information players have about
each other. The categories used in this book are:
1. Perfect information: each player knows where they are in the game and who they are playing.
2. Incomplete information: a pseudo-player called ‘nature’ or ‘chance’ moves in a random way that
is not clearly observed by all or some of the players.
3. Asymmetric information: not all the players observe the chance move. Not all players have the
same information. Instead some player has private information.
When information is not perfect there is uncertainty in one or more of the players’ minds about
where they are in a game or who they are playing. Where risk is involved decision makers need to
incorporate the relevant probabilities into their decision making.
Chapter 2 Moving together
2.1 Dominant-strategy equilibrium
In a dominant-strategy equilibrium every player in the game chooses their dominant strategy. A
dominant strategy is a strategy that is a best response to all the possible strategy choices of all the
other players. A game will only have a dominant-strategy equilibrium
if all the players have a dominant strategy. Some example of games
are:
Pub Managers Game: to see if the game has a dominant-strategy
equilibrium we need to check whether both players have a
dominant strategy. If the Queen’s Head manager makes the
special offer his pay-off is either 10 or 18. If the manager of the Queen’s Head doesn’t make the
, offer then his pay-off is either 4 or 7. Thus making the offer is a dominant strategy for him (10>4
and 18>7). Similar the domanat strategy of the manager of the King’s Head is the special offer
(20>14 and 8>6).
Labour market legislation: the pay-offs in Matrix 2.2 are expected gains in millions of votes. In
the legislation game the dominant strategy of both governments is to introduce the legislation.
Port access: As only the ranking of the pay-offs to the players is important and it is difficult to
conceive of meaningful numerical equivalents for the utilities of the two governments, the pay-
offs are delineated as letters. Refusing access is therefore a dominant strategy for New Zealand.
The alliance is a dominant strategy for the USA. The dominant-strategy equilibrium of this game
is therefore {maintain the alliance, refuse access}.
Dominant strategies & Dominant-strategy equilibrium
Strongly dominant strategy: in a two-player game the pay-offs to a player from choosing a
strongly dominant strategy are higher than those from choosing any other strategy in response
to any strategy the other player chooses.
Weakly dominant strategy: in a two-player game the pay-offs to a player from choosing a weakly
dominant strategy are:
o At least as high as those from choosing any other strategy in response to any strategy the
other player chooses and
o Higher than those from choosing any other strategy in response to at least one strategy
of the other player
Strong dominant-strategy equilibrium: a combination of strongly dominant strategies; in a two-
player game a pair of strategies that for each player are strictly best responses to all of the
strategies of the other player.
Weak dominant-strategy equilibrium: combination of dominant strategies where some or all of
the strategies are only weakly dominant.
2.2 Iterated-dominance equilibrium
Many games do not have a dominant-strategy equilibrium. In this case we can look for an iterated-
dominance equilibrium. If one of the players in a two-player game has a dominant strategy then even
if the other player doesn’t the game may still have an iterated-dominance equilibrium. More
generally in a two person game an iterated-dominance equilibrium is a strategy combination where
for at least one player their equilibrium strategy is:
As good any other strategy and better than some in response to all the non-dominated strategies
of the other player;
A best response to the equilibrium strategy of the other player.
Overview of the book
Chapter 1: basic ideas and concepts underlying game theory
Chapter 2: simultaneous- or hidden-move games are analysed and the dominant strategy and Nash
equilibrium concepts are defined.
Chapter 3: prisoners’ dilemma
Chapter 4: how sequential decision making can be modelled using game theory and extensive forms
Chapters 6: Nash equilibrium concept is extended to incorporate randomising or mixed strategies
Chapter 8: more realism is incorporated by allowing for the possibility that people play some games
more than once
Chapter 9: the methodology used to analyse dynamic games in Chapter 4 is applied to strategic
bargaining problems
--------------------------------------------------------------------------------------------------------------------------------------
Chapter 1 Game Theory Toolbox
1.1 The idea of game theory
Game theory is:
An intellectual framework
for examining what various parties to a decision should do
given their possession of inadequate information and different objectives
Game theory is 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). In these circumstances
the plans or strategies of the individuals concerned will be dependent on expectations about what
the others are doing. Thus individuals in these kinds of situations are not making decisions in
isolation, instead their decision making is interdependently related.
Strategic game: a scenario or situation where for two or more individuals their choice of action or
behaviour has an impact on the other (or others).
Strategic interdependence: individuals’ decisions, their choices about actions, impact on each
other and therefore their decision making is interdependently related.
Player: a participant in a strategic game.
Strategy: a player’s plan of action for the game.
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 knowledge of
the game, their possible moves or actions 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. 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. Payoffs can be
described as e.g. I like chocolate (A) more than pizza (B), but this is simpler A>B. 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. 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.
,Games are often characterised 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.
Pay-offs, equilibrium and rationality:
Pay-off: measures how well the player does in a possible outcome of a game. Pay-offs are
measured in either material rewards (money) or in utility that a player derives from a particular
outcome of a game.
Utility: a subjective measure of a player’s satisfaction, pleasure or the value they derive from a
particular outcome of a game.
Equilibrium strategy: a ‘best’ strategy for a player in that it gives the player his or her highest pay-
off given the strategy choices of all the players.
Equilibrium in a game: a combination of players’ strategies that are a best response to each
other.
Rational play: players choose strategies with the aim of maximising their pay-offs.
1.3 Simultaneous-move games
In these games players make moves at the same time or, what amounts to the same thing, their
moves are unseen by the other player. There are 3 types of games:
1. Hide and seek: e.g. a hide and seek game in a house. If Tim finds Robina in the time allotted he
wins €50, otherwise Robina wins the €50. The pay-off matrix is shown below. It shows all the
possible pay-offs of the players that result from all their possible strategy combinations. In each
cell the pay-off of the player whose actions are designated by the rows of the matrix are written
first (Robina). The pay-offs of the player whose actions are denoted in the columns are written
second (Tim).
2. Pub manager’s game: both managers are simultaneously considering introducing a special offer
to their customers by cutting the price of their premium beer.
3. Penalty taking: in this game 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. Notice that in the cells of Matrix the pay-offs
always add to the constant sum 10 since if one player’s pay-off is 10 the other’s is zero.
Conclusions of the games above:
Diametrically opposed: the first and third games as if one wins the other effectively loses. Games
like this are games of pure conflict.
, Constant-sum games: games in which the sum of the players’ pay-offs is a constant. If the
constant sum is zero the game is a zero-sum game. Constant-sum games are games of pure
conflict; one player’s gain is the other’s loss. E.g. Hide-and-seek and the penalty-taking game.
Mixed-motive games: A mix of pure strategies determined by a randomisation procedure. 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
1.4 Sequential-move or dynamic games
In sequential-move games players make moves in some sort of order. One of the players moves first
and another sees the first player’s move before deciding how to respond. It is not always easy to do
this using pay-off matrices and therefore sequential games are usually analysed using game trees.
The payoffs are written in the same order as the players’ moves, i.e. the pay-off of the player who
moves first (A) 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.
1.6 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. In a non-
cooperative games agreements cant being enforced.
1.7 N-player games
N is the number of players in the game. If a game has two players then it is a 2- player game. But if
there are more than two players then the game is an N-player game where N is greater than 2.
1.8 Information
The equilibrium strategies of the players will depend on what kind of information players have about
each other. The categories used in this book are:
1. Perfect information: each player knows where they are in the game and who they are playing.
2. Incomplete information: a pseudo-player called ‘nature’ or ‘chance’ moves in a random way that
is not clearly observed by all or some of the players.
3. Asymmetric information: not all the players observe the chance move. Not all players have the
same information. Instead some player has private information.
When information is not perfect there is uncertainty in one or more of the players’ minds about
where they are in a game or who they are playing. Where risk is involved decision makers need to
incorporate the relevant probabilities into their decision making.
Chapter 2 Moving together
2.1 Dominant-strategy equilibrium
In a dominant-strategy equilibrium every player in the game chooses their dominant strategy. A
dominant strategy is a strategy that is a best response to all the possible strategy choices of all the
other players. A game will only have a dominant-strategy equilibrium
if all the players have a dominant strategy. Some example of games
are:
Pub Managers Game: to see if the game has a dominant-strategy
equilibrium we need to check whether both players have a
dominant strategy. If the Queen’s Head manager makes the
special offer his pay-off is either 10 or 18. If the manager of the Queen’s Head doesn’t make the
, offer then his pay-off is either 4 or 7. Thus making the offer is a dominant strategy for him (10>4
and 18>7). Similar the domanat strategy of the manager of the King’s Head is the special offer
(20>14 and 8>6).
Labour market legislation: the pay-offs in Matrix 2.2 are expected gains in millions of votes. In
the legislation game the dominant strategy of both governments is to introduce the legislation.
Port access: As only the ranking of the pay-offs to the players is important and it is difficult to
conceive of meaningful numerical equivalents for the utilities of the two governments, the pay-
offs are delineated as letters. Refusing access is therefore a dominant strategy for New Zealand.
The alliance is a dominant strategy for the USA. The dominant-strategy equilibrium of this game
is therefore {maintain the alliance, refuse access}.
Dominant strategies & Dominant-strategy equilibrium
Strongly dominant strategy: in a two-player game the pay-offs to a player from choosing a
strongly dominant strategy are higher than those from choosing any other strategy in response
to any strategy the other player chooses.
Weakly dominant strategy: in a two-player game the pay-offs to a player from choosing a weakly
dominant strategy are:
o At least as high as those from choosing any other strategy in response to any strategy the
other player chooses and
o Higher than those from choosing any other strategy in response to at least one strategy
of the other player
Strong dominant-strategy equilibrium: a combination of strongly dominant strategies; in a two-
player game a pair of strategies that for each player are strictly best responses to all of the
strategies of the other player.
Weak dominant-strategy equilibrium: combination of dominant strategies where some or all of
the strategies are only weakly dominant.
2.2 Iterated-dominance equilibrium
Many games do not have a dominant-strategy equilibrium. In this case we can look for an iterated-
dominance equilibrium. If one of the players in a two-player game has a dominant strategy then even
if the other player doesn’t the game may still have an iterated-dominance equilibrium. More
generally in a two person game an iterated-dominance equilibrium is a strategy combination where
for at least one player their equilibrium strategy is:
As good any other strategy and better than some in response to all the non-dominated strategies
of the other player;
A best response to the equilibrium strategy of the other player.
