A guide to game theory by Fiona Carmichael
Chapter 1 – Game theory toolbox
Game theory: a technique used to analyse situatons here for t o or more individuals (or
insttutons) the outcome of an acton by one of them depends not only on the
partcular acton taken by that individual but also on the actons taken by the other
(or others)
The plans or strategies of individuals are depended on expectatons about hat the others are
doing
Individuals in these kind of situatons are not making decisions in isolatonn instead their decision
making is independently related
o This is called strategic interdependence and such situatons are commonly kno n as
games of strategyn or simply gamesn hile the partcipants in such games are referred to
as players
o Strategy is a players plan of acton for the game
In strategic games the actons of one individual or group impact on others andn cruciallyn the
individuals involved are a are of this
Because players in a game are conscious that the outcomes of their actons are aaected by and aaect
others they need to take into account the possible actons of these other individuals hen they
themselves make decisions.
When they have limited info about the others they have to make conjectures about hat think
they ill do
o These kind of thought processes consttute strategic to understand hat is going on and
make predictons about likely outcomes
Making plans in strategic situatons requires thinking carefully before you actn taking into account
hat you think the people you are interactng ith are also thinking about and planning.
- Game theory gives sharp analytc tools in order to explain behaviour and predict outcomes in
strategic situatons
Describing strategic games
First it is necessary to defne the boundaries of the strategic game under constructon. Games are
defned in terms of their rules. The rules of a game incorporate informaton about the playerss
identty and their kno ledge of the gamen their possible moves or actons and their pay-oas.
- The rules of a game describe in detail ho one playerss behaviour impacts on other playerss pay-
oas.
- A player can be any kind of thinking entty that is generally assumed to act ratonally and is
involved in a strategic game ith one or more other players
Pay oa measures ho ell the player does in a possible outcome of a game. Pay-oas are
measured in terms of either material re ards such as money or in terms of the utlity
that a player derives from a partcular outcome of a game
You can measure ith numbers but also rank ith for example An B etc.
Utlity a subjectve measure of a playerss satsfactonn pleasure or the value they derive from
a partcular outcome of a game
Utls: units of utlity
Ratonal individuals are assumed to prefer more utlity to less and therefore in a strategic game a
pay-oa that represents more utlity ill be preferred to one that represents less.
Not that hile this ill al ays be true about levels of satsfacton or pleasure it ill not al ays be
the case hen e are talking about quanttes of material goods like chocolate (can eat to much)
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,Equilibrium strategy a ‘bests strategy for a player in that it gives the player his or her highest pay-
oa given the strategy choices of all the players
Equilibrium in a game a combinaton of playerss strategies that are a best response to each other
Ratonal play players choose strategies ith the aim of maximising their pay-oas
Games in hich players move at the same tme or their moves are hidden are called simultaneous-
move or statc games. Games in hich the players move in some kind of predetermined order are
called sequental-move or dynamic games.
Simultaneous-move games
1. Hide-and-seek
This is a hidden-move game and is analysed using the pay-oa matrix. The interests of the players
are diametrically opposed; if one ins the other eaectvely loses; games of pure confict.
o Constant sum game
o The pay-oa matrix sho s ho gets the re ardn it is necessary to put them in one matrix.
2. A pub managerss game
This is a simultaneous-move game and is analysed using the pay-oa matrix. This is a mixed-
motve game. In such games there ill be mutually benefcial or mutually harmful outcomes so
that there are shared objectves (most ofen).
o You are going to look hat the re ard ill be if only one of the t o changes somethingn
if you both change or if no one changes. Put this also in one matrix.
3. Penalty-taking game
This is a simultaneous-move game and is analysed using the pay-oa matrix. The interests of the
players are diametrically opposed; if one ins the other eaectvely loses; games of pure confict.
o Constant sum game
o In this gamen pay-oas can not be measured in terms of money; they are represented in
terms of levels of subjectve satsfacton or utlity
o You assign scores to the utlity that people feel. It could be that one fnds inning more
important than the other onen so you could assign diaerent scores. Remember that the
scores are not directly comparable so this ould really be an unnecessary complicaton
o In order to construct the pay-oa matrix that corresponds to these pay-oas e need to
make some additonal assumptons:
The striker al ays kicks the ball on target (so he either scores or the goalkeeper
makes a save
Simplify the strategy (striker can only kick lefn right or in the middlen goalkeeper
can only usen lef or right hand or stay on the ground)n if the goalkeeper mirrors
the acton of the striker he ins
o Again make the matrix ith all the outcomes in it
You can also use non-nummerical pay-oas
Constant-sum and zero-sum games games in which the sum of the players’ pay-ofs is a constant.
If the constant sum is zero the game is a zero-sum game.
Constant-sum games are games of pure confictt one player’s
gain is the other’s loss
Pure confict onst al ays be constant-sum games although they can usually be represented in
this ay
In the penalty-taking game choosing lefn right or middle is a pure strategy. It is possible that
someone does choose hen running to the ball for examplen or by rolling the dice. In this case there
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, ould be a 1/3 change that one opton is chosen. These are strategies that mix up a playerss pure
strategies and are therefore called mixed strategies (i.e. a mix of pure strategies determined by a
randomisaton procedure).
Sequential-move or dynamic games
In sequental-move games players make moves in some sort of order. This means one player moves
frst and the other player of players see the frst playerss 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.
Sequental games are usually analysed using game trees or extensive forms
o The decision tree sho s at the end the numbers for the frst company and then the
second one.
Games that are only played once by the same players are called one-shotn single-stage or unrepeated
games. Games that are played by the same players more than once are kno n as repeatedn mult-
stage or n-stage games here 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 repetton or stage of the game. These kinds of
strategies are called meta-strategies.
- The penalty game could be played repeatedlyn and rite it do ns as anbncndne etc.
There are cooperatve and non-cooperatve games. The most games are in a technical sense not
cooperatve.
N-player games: n is the number of players in the game
The greater the number of players involved in a game the more complex it is likely to be.
Information:
- Perfect informaton if info if perfect each player knows where they are in the game and
who they are playing
- Incomplete informaton if info is incomplete then a pseudo-player called ‘nature’ of ‘chance’
moves in a random way that is not clearly observed by all or some of
the players
- Asymmetric informaton not all players have the same informatonn instead some player has
private informaton
hether the situaton is strategic or notn here risk is involved decision makers need to incorporate
the relevant probabilites into their decision making. They do this by forming expectatons about
likely outcomes and ratonal decision makers are assumed to choose in order to maximise their
expected pay-oa. This is an average of all the possible pay-oas corresponding to a given choice. It is
calculated by multplying (or eightng) each pay-oa by the probability that it ill occur.
Expected utlity is potentally the more useful measure as it can incorporate peopless diaerent
attitudes to risk.
Summary on p.18
Chapter 2 – Moving together
Dominant strategy equilibrium every player in the game chooses their dominant strategy
A strategy that is a best response to all the possible strategy choices of all the other players
A game ill only have a dominant-strategy equilibrium if all the players have a dominant strategy
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, This means that choosing to do something is preferred over not doing somethingn because both
opportunites give you more proft (you do something and the other party toon or you do something
and the other party doesnst gives you more money then hen you donst do anything).
When choosing the doing something strategy for both e can speak of the dominant strategy
equilibrium you rite it do n as {special oaern special oaerr (same as in a tablen but in ords).
In shortn in a dominant strategy equilibrium all players choose their dominant strategies to maximize
pay-oas and there is no ratonal reason to believe that the player ould choose some other opton.
- Strongly dominant strategy when one pay-of of a strategy is way higher than another
o A combinaton of strongly dominant strategies; in t o-player game a pair of strategies
that for each player are strictly best responses to all of the strategies of the other player
- Weakly dominant strategy when the pay-of is (1 least as high as those from choosing
any other strategy in response to any strategy the other
player chooses and 21 higher than those from choosing any
other strategy in response to at least one strategy of the
other player
o Combinaton of dominant strategies here some or all of the strategies are only eakly
dominant
o The pay-oa to the player from choosing that strategy is at least as good as any other
strategy and beter than some in response to hatever strategy the other player picks
If a player in a game has a dominant strategy all their other strategies are dominated strategies.
Watch out!! Dominant equilibrium does not have to mean that both parties choose the same
strategy!!
Iterated-dominance equilibrium an equilibrium found by deletng strongly or eakly
dominated strategies untl only one pair of strategies remains
If one of the player either have a strongly or eakly dominant strategy
An additonal requirement hen the player ith the dominant strategy has a choice of only t o
strategies is that the other player has a best response to the dominant strategy of the frst player
(this second player doesnst has to have a dominant strategy)
In shortn in a t o-person game an iterated-dominance equilibrium is a strategy combinaton here
for at least one player their equilibrium strategy (1) is as good as any other strategy and beter than
some in response to all the non-dominated strategies of the other player and (2) is a best response to
the equilibrium strategy of the other player.
For the other playern their equilibrium strategy is a best response to the equilibrium strategy
of the frst player.
So you delete the dominated strategies of both and then you kno hich strategy to choose. You
rite it do n as {opton person 1n opton person 2r.
Watch out! Look really carefully if they mean dominated or dominant
You can make a diaerence bet een eak iterated-dominance equilibrium ( hen something is
eakly dominated) and strong iterated-dominance equilibrium.
There can be situatons here there is more than one iterated-dominance equilibrium. There is more
than one eakly dominated strategies and the order of deleton maters. This is something to be
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