i\
\'
1: GAMES IN STRATEGIC FORM
\4ARTIN CRIPPS
The purpose of this topic is to tetrch 1-ou u'hat a garne in strategic form is and how
economists analyze them.
1. Ganros rN SrRATtrGrc l-oRNr: DpscRrprroN AND Erarrpr,ps
1.1. Defining a Game in Strategic Forrn. A game in strategic form consists of sevelal
elernents. Here we are quite precise about what must be included in the description. Thet'e
rnust be the following three things:
(1) A list of pla-vers; i,:7,2,...,1. This list can be finite or infinite. Here the narnes
of the pla\.ers are just nurnbers but they can be any entity you are interested in
strrd5-irrg.
Exanrples:
(a) In Rock-Paper-Scissors there are two players i :Frcd, Dais1,'.
(b) In an oligopoly with 5 fi,rms there are n, : 5 p1a1.ers i :Ford. Honda. Toyottr,
VW, Fiat, G\4.
(c) In bargaining there is a buyer and a seller. so the list of plavers is I :buyer,
seller.
(2) A description of all the actions each pla1'er can take. Tire set/1ist of possible
actions for player i is usually called their pure strategies b1- game theorists. Wc rnill
represent this b1'the set S; arrd we write a t1.picttl action as s; € S;.
Examples:
(a) In Rock-Paper-Scissors, a player has thlee pure strategies: so
s; € {Rock, Paper, Scissors} : S;, fol i : 7.2.
(b) In an oligopolv,each firm. z. will choose a nolr-negative quantity of output to
produce, 0 I q, the set ,S1 is the set of all positive output quantities.
(c) In bargaining. the bu1.s1 proposes a price 0 < p ( 1 and the seller makes a
proposal too so the strategy sets are just intervals of prices: Sb.,y", : {0 <
p<1),Sseller:{0<p<1}
When you want to w'rite ali the actions that were taken by all of the players
you will write a list or vector ,: (rr,...,sr). This is called an action or strategy
profile. The set of all possible strategy profiles (all possible plays of the game) is
written as S , where s : (sr.. ... s7) e S.
Examples:
1
, 2 MARTIN CRIPPS
(a) In Rock-Paper-Scissors: arr exanrple of a profile is ,s : (Scissors, Paper) here
player 1 does ,S a.nd 2 does P. The set of all possible profiles ,S has 9 elements.
(b) In oligopoly: a pr-oflle is a list of quantrties for every firrn each firm, f . That is
** d alt a list (q1 ,Q2,...,Qr). The set of all possible profiles is Rf .
{2"9*\\il1 (c) Bargaining: a pro{ile is tr,vo prices (pr,,pr) and the set of all possible profiles is
a/.h,Ovta.t 9 [0, r]'.
(3) The final element is the payoffs (or utilitl' or profit) the plavers get frorn their
a"xtrw actions or pure strategies. We can write this as tr utilityfpayoff function ur(s) :
= @fuy*rt ui(sr,..., sr), that determines player i?s pavoff at every possible play of the game.
The pa1'offs are usually deterrnined by the rules of the garne or economic phenom-
enon you are stuciying.
Examples:
(a) (Rock Paper Scissors): Recall there are tr,vo pl:r1'ers zl : 1,2 and each has three
actions ,9r : {R,P,,S} and the payoffs can be represented in a table:
RPS
R 0,0 -1,1 1,-1
P 1,-1 0,0 -1,1
S -1,1 1,-1 0,0
(b) (Oligopolr'r'r'here firurs choose quantities): Recall that the firms'names are
z : 1.2.....n. The firurs? actiotrs are ontpr-rts e; ) 0. The firms, payoffs are
their plofits. u-hich depends on the clenrtrnd. Plice u.ill depend on total output
so rve r,r'ite P(qr+e,z1,.. tA"). (FoI'exrlmple P:50-q,
-e2...- ar) profit
also depetrcls on a firm's costs. ',r'e assllnre these only depend on their output
c(q,). Hence
Profit of F-irm z : Revenue - Costs
: Or-rtpnt of I x Price - Costs of i
: qrP(qr ] qz + .. * a") - c(qi)
This completes our initial description of a game in strategic fbrm. We now want to ailow
for the possibility that players act randomll.. We don't necessarily think that players are
genuinely randomising. But it may be a very good description of what the players think
the otliers are doing. You may know exactly how yoll're going to play Paper-Scissors-Rock,
but I don't. From my point of view your action looks random. We call random actions
Ilfi,red Strategies. On mixed action for a player is one probability distribution. so the set
of all the player's mixed strategies is the set of all probabilitv distributions on their plre
actions. We write the rnixed strategy of player f as o;. We will write the proflle of mixed
strategies for all plavers as o : (or,...,o1).
Exanrplel: Inpaper.scissorsrock,S.:{R,P,,5}and oi:(p,q,l*p-q) wherepisthc
probability R is played. q is the probability P is played and 1-p g is the probability S
is played.
, 1: GA\IES IN STRATEGIC FOR\I 3
Example 2: When firms choose quantities S, : [0,
-) is the set of possible quantities. A
random choice ofa quantity can be represented by a probability distribution over the set of
positive numbers. of descri bing such a distribution is to *'rite down its cumulative
distribution function F(r) :: Pr(Firm's outputs less than c-ir equal to x).
Payoffs from Mixed Actions: A player's payoff when mixed actions are played is an expec-
tation (or average) taken over all the they may get multiplied by the probability
they get them. This expectation is
Exarnples of payoffs from mixed actions (Rock Paper Scissors): Suppose in this game yorl
are the colurnn player and you believe the rorv plaver u'ill play action R with probability
p, P with probability q ancl S with probabilitr- 1 - p q. \bu are interested in your payoff
from action P. Below we have rn''r'itteu out tire pa1'offs and emphrrsisecl the relevant numbers
for the column plaver.
R p S
I) 0.0 -1.1 1.-1
() 1,- 1 0.0 -1.1
7 p-ct -1,1 1,-1 0,0
If she plays P she expects to get 1 r,vith probability p, 0 r,vith probability q and -1 with
probability 7 -p- q. Thus on average she expects to get
px1*qx0*(1 -p-s) x(-1) :2p-tq*L.
If we repeat these calculations for each column we can work out what each action will give
irer in expectation
RPS
p 0,0 -1,1 1,-
q 1,- 1 0,0 -1, 1
l-p-q -1,1 1,-1 0
-n_r q -fcl- q-p
2. Dor,rrNaucp
2.1.Strict Dominance.
Definition L. A mired strate.g'y o; strictly d,orn'inates th,e p'ure acti,on s'n for playeri,'r,f and
only i,f, playeri's payoff when she plays oi and the other players play actions s-1 iis strictl,y
h'igh,er th,an her payoff frorn st, against s- i for any actions s-; the others may plu,y:
ut(ot,r-,) > u;(sl. s-;). Vs-i.
Consider the following game (we onll' put in the rou, plaver's payoffs as that is all that
matters for now) .
LR
T 30
N4 03
B 11
, 4 TIARTIN CRIPPS
And consider tlvo strategies for the lorv plaver: plal' the first tr,vo rows with equal probabilitl.
oi: (712,112,0), or play the bottorn row sr : B. \Ve can."vlite the expected pa1'offs to
these strategies in the follo"r,ing wa)..
L R L R
, 3i)
1
L) 30
, 03 03
1
0
0 11 1 11
1
1
From this vou ctln see that if the column player plzr1's I the stra,tegr.' o, gives the row player
the pavoff of j but the strategy s, gives the rolv pial-el the pn1.6ff 1. Ancl. if the column
plaver piays R strategy o; also gives the row piayer the par-off of j] but the strategy s, gives
the row plaver the payoff 1. Thus the strategy oi al$ravs does better than the strategy
s;. To describe this we say si is strr,ctly dom'inated and ',r'e u,ou.lcl never expect a rational
player to play this.
However, eliminating strictll'dominated actions cau allou. us to make strong predictions
about what actions the players will use. Consider tlie fb11ou-irrg ga1r1e. . .
LR
U (8,10) (-100,e)
D (/.rr) (6 5)
First observe that -R is strictll'dorninated b1,- Z for the cohirnn plal.er. So a rational column
player will nevel plal' /?. If the row plal,er krro'uvs this then they should play t/ getting
B rather than 7. So w'e predict (Lf , L) as the olrtcome of this garre. But, some types
of players rnay be very worried about the 100. If r-or-r hir<l some doubts about coLurnn
player's rationality would you be willing to plar- L:?
2.2. Weak Dominance. The riotion of u.eak domination does not require pltqers to
strictly prefer one strategy to another, it is enough to u,eakh, pre{'er one to the other.
Definition 2. A mired strategy oi'ueakly dornirtates the pu,re acti,on s, € S; for player i,
i'f and only if. playing o;'is at least as good as playing si whatluer the other plauers do:
rr1(o;. s-,) ). ui(.s,..s-;). Vs-i.
L R
T (1 1) 0, o)
\{ ( 0,0) 0 ) o)
n
T) ( 0,0) 0,0)
T weakly dominates NtI for the rou, plaver. because T is better than N{ if column plays L
and T is no worsc than M is ccilurnn plavs R.
\'
1: GAMES IN STRATEGIC FORM
\4ARTIN CRIPPS
The purpose of this topic is to tetrch 1-ou u'hat a garne in strategic form is and how
economists analyze them.
1. Ganros rN SrRATtrGrc l-oRNr: DpscRrprroN AND Erarrpr,ps
1.1. Defining a Game in Strategic Forrn. A game in strategic form consists of sevelal
elernents. Here we are quite precise about what must be included in the description. Thet'e
rnust be the following three things:
(1) A list of pla-vers; i,:7,2,...,1. This list can be finite or infinite. Here the narnes
of the pla\.ers are just nurnbers but they can be any entity you are interested in
strrd5-irrg.
Exanrples:
(a) In Rock-Paper-Scissors there are two players i :Frcd, Dais1,'.
(b) In an oligopoly with 5 fi,rms there are n, : 5 p1a1.ers i :Ford. Honda. Toyottr,
VW, Fiat, G\4.
(c) In bargaining there is a buyer and a seller. so the list of plavers is I :buyer,
seller.
(2) A description of all the actions each pla1'er can take. Tire set/1ist of possible
actions for player i is usually called their pure strategies b1- game theorists. Wc rnill
represent this b1'the set S; arrd we write a t1.picttl action as s; € S;.
Examples:
(a) In Rock-Paper-Scissors, a player has thlee pure strategies: so
s; € {Rock, Paper, Scissors} : S;, fol i : 7.2.
(b) In an oligopolv,each firm. z. will choose a nolr-negative quantity of output to
produce, 0 I q, the set ,S1 is the set of all positive output quantities.
(c) In bargaining. the bu1.s1 proposes a price 0 < p ( 1 and the seller makes a
proposal too so the strategy sets are just intervals of prices: Sb.,y", : {0 <
p<1),Sseller:{0<p<1}
When you want to w'rite ali the actions that were taken by all of the players
you will write a list or vector ,: (rr,...,sr). This is called an action or strategy
profile. The set of all possible strategy profiles (all possible plays of the game) is
written as S , where s : (sr.. ... s7) e S.
Examples:
1
, 2 MARTIN CRIPPS
(a) In Rock-Paper-Scissors: arr exanrple of a profile is ,s : (Scissors, Paper) here
player 1 does ,S a.nd 2 does P. The set of all possible profiles ,S has 9 elements.
(b) In oligopoly: a pr-oflle is a list of quantrties for every firrn each firm, f . That is
** d alt a list (q1 ,Q2,...,Qr). The set of all possible profiles is Rf .
{2"9*\\il1 (c) Bargaining: a pro{ile is tr,vo prices (pr,,pr) and the set of all possible profiles is
a/.h,Ovta.t 9 [0, r]'.
(3) The final element is the payoffs (or utilitl' or profit) the plavers get frorn their
a"xtrw actions or pure strategies. We can write this as tr utilityfpayoff function ur(s) :
= @fuy*rt ui(sr,..., sr), that determines player i?s pavoff at every possible play of the game.
The pa1'offs are usually deterrnined by the rules of the garne or economic phenom-
enon you are stuciying.
Examples:
(a) (Rock Paper Scissors): Recall there are tr,vo pl:r1'ers zl : 1,2 and each has three
actions ,9r : {R,P,,S} and the payoffs can be represented in a table:
RPS
R 0,0 -1,1 1,-1
P 1,-1 0,0 -1,1
S -1,1 1,-1 0,0
(b) (Oligopolr'r'r'here firurs choose quantities): Recall that the firms'names are
z : 1.2.....n. The firurs? actiotrs are ontpr-rts e; ) 0. The firms, payoffs are
their plofits. u-hich depends on the clenrtrnd. Plice u.ill depend on total output
so rve r,r'ite P(qr+e,z1,.. tA"). (FoI'exrlmple P:50-q,
-e2...- ar) profit
also depetrcls on a firm's costs. ',r'e assllnre these only depend on their output
c(q,). Hence
Profit of F-irm z : Revenue - Costs
: Or-rtpnt of I x Price - Costs of i
: qrP(qr ] qz + .. * a") - c(qi)
This completes our initial description of a game in strategic fbrm. We now want to ailow
for the possibility that players act randomll.. We don't necessarily think that players are
genuinely randomising. But it may be a very good description of what the players think
the otliers are doing. You may know exactly how yoll're going to play Paper-Scissors-Rock,
but I don't. From my point of view your action looks random. We call random actions
Ilfi,red Strategies. On mixed action for a player is one probability distribution. so the set
of all the player's mixed strategies is the set of all probabilitv distributions on their plre
actions. We write the rnixed strategy of player f as o;. We will write the proflle of mixed
strategies for all plavers as o : (or,...,o1).
Exanrplel: Inpaper.scissorsrock,S.:{R,P,,5}and oi:(p,q,l*p-q) wherepisthc
probability R is played. q is the probability P is played and 1-p g is the probability S
is played.
, 1: GA\IES IN STRATEGIC FOR\I 3
Example 2: When firms choose quantities S, : [0,
-) is the set of possible quantities. A
random choice ofa quantity can be represented by a probability distribution over the set of
positive numbers. of descri bing such a distribution is to *'rite down its cumulative
distribution function F(r) :: Pr(Firm's outputs less than c-ir equal to x).
Payoffs from Mixed Actions: A player's payoff when mixed actions are played is an expec-
tation (or average) taken over all the they may get multiplied by the probability
they get them. This expectation is
Exarnples of payoffs from mixed actions (Rock Paper Scissors): Suppose in this game yorl
are the colurnn player and you believe the rorv plaver u'ill play action R with probability
p, P with probability q ancl S with probabilitr- 1 - p q. \bu are interested in your payoff
from action P. Below we have rn''r'itteu out tire pa1'offs and emphrrsisecl the relevant numbers
for the column plaver.
R p S
I) 0.0 -1.1 1.-1
() 1,- 1 0.0 -1.1
7 p-ct -1,1 1,-1 0,0
If she plays P she expects to get 1 r,vith probability p, 0 r,vith probability q and -1 with
probability 7 -p- q. Thus on average she expects to get
px1*qx0*(1 -p-s) x(-1) :2p-tq*L.
If we repeat these calculations for each column we can work out what each action will give
irer in expectation
RPS
p 0,0 -1,1 1,-
q 1,- 1 0,0 -1, 1
l-p-q -1,1 1,-1 0
-n_r q -fcl- q-p
2. Dor,rrNaucp
2.1.Strict Dominance.
Definition L. A mired strate.g'y o; strictly d,orn'inates th,e p'ure acti,on s'n for playeri,'r,f and
only i,f, playeri's payoff when she plays oi and the other players play actions s-1 iis strictl,y
h'igh,er th,an her payoff frorn st, against s- i for any actions s-; the others may plu,y:
ut(ot,r-,) > u;(sl. s-;). Vs-i.
Consider the following game (we onll' put in the rou, plaver's payoffs as that is all that
matters for now) .
LR
T 30
N4 03
B 11
, 4 TIARTIN CRIPPS
And consider tlvo strategies for the lorv plaver: plal' the first tr,vo rows with equal probabilitl.
oi: (712,112,0), or play the bottorn row sr : B. \Ve can."vlite the expected pa1'offs to
these strategies in the follo"r,ing wa)..
L R L R
, 3i)
1
L) 30
, 03 03
1
0
0 11 1 11
1
1
From this vou ctln see that if the column player plzr1's I the stra,tegr.' o, gives the row player
the pavoff of j but the strategy s, gives the rolv pial-el the pn1.6ff 1. Ancl. if the column
plaver piays R strategy o; also gives the row piayer the par-off of j] but the strategy s, gives
the row plaver the payoff 1. Thus the strategy oi al$ravs does better than the strategy
s;. To describe this we say si is strr,ctly dom'inated and ',r'e u,ou.lcl never expect a rational
player to play this.
However, eliminating strictll'dominated actions cau allou. us to make strong predictions
about what actions the players will use. Consider tlie fb11ou-irrg ga1r1e. . .
LR
U (8,10) (-100,e)
D (/.rr) (6 5)
First observe that -R is strictll'dorninated b1,- Z for the cohirnn plal.er. So a rational column
player will nevel plal' /?. If the row plal,er krro'uvs this then they should play t/ getting
B rather than 7. So w'e predict (Lf , L) as the olrtcome of this garre. But, some types
of players rnay be very worried about the 100. If r-or-r hir<l some doubts about coLurnn
player's rationality would you be willing to plar- L:?
2.2. Weak Dominance. The riotion of u.eak domination does not require pltqers to
strictly prefer one strategy to another, it is enough to u,eakh, pre{'er one to the other.
Definition 2. A mired strategy oi'ueakly dornirtates the pu,re acti,on s, € S; for player i,
i'f and only if. playing o;'is at least as good as playing si whatluer the other plauers do:
rr1(o;. s-,) ). ui(.s,..s-;). Vs-i.
L R
T (1 1) 0, o)
\{ ( 0,0) 0 ) o)
n
T) ( 0,0) 0,0)
T weakly dominates NtI for the rou, plaver. because T is better than N{ if column plays L
and T is no worsc than M is ccilurnn plavs R.
