Listofplayers
Payoffs
Purestrategies
1: GAMES IN STRATEGIC FORM
MARTIN CRIPPS
The purpose of this topic is to teach you what a game in strategic form is and how
economists analyze them.
1. Games in Strategic Form: Description and Examples
1.1. Defining a Game in Strategic Form. A game in strategic form consists of several
elements. Here we are quite precise about what must be included in the description. There
must be the following three things:
(1) A list of players; i = 1, 2, . . . , I. This list can be finite or infinite. Here the names
of the players are just numbers but they can be any entity you are interested in
studying.
Examples:
(a) In Rock-Paper-Scissors there are two players i =Fred, Daisy.
(b) In an oligopoly with 5 firms there are n = 5 players i =Ford, Honda, Toyota,
VW, Fiat, GM.
(c) In bargaining there is a buyer and a seller, so the list of players is i =buyer,
seller.
(2) A description of all the actions each player can take. The set/list of possible
actions for player i is usually called their pure strategies by game theorists. We will
represent this by the set Si and we write a typical action as si 2 Si .
Examples:
(a) In Rock-Paper-Scissors, a player has three pure strategies: so
si 2 {Rock, Paper, Scissors} = Si , for i = 1, 2.
(b) In an oligopoly, each firm, i, will choose a non-negative quantity of output to
produce, 0 qi the set Si is the set of all positive output quantities.
(c) In bargaining, the buyer proposes a price 0 p 1 and the seller makes a
proposal too so the strategy sets are just intervals of prices: Sbuyer = {0
p 1}, Sseller = {0 p 1}.
When you want to write all the actions that were taken by all of the players
you will write a list or vector s = (s1 , . . . , sI ). 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 = (s1 , . . . , sI ) 2 S.
Examples:
1
,2 MARTIN CRIPPS
(a) In Rock-Paper-Scissors: an example of a profile is s = (Scissors, Paper) here
player 1 does S and 2 does P . The set of all possible profiles S has 9 elements.
(b) In oligopoly: a profile is a list of quantities for every firm each firm, i. That is
a list (q1 , q2 , . . . , qn ). The set of all possible profiles is Rn+ .
(c) Bargaining: a profile is two prices (pb , ps ) and the set of all possible profiles is
[0, 1]2 .
(3) The final element is the payo↵s (or utility or profit) the players get from their
actions or pure strategies. We can write this as a utility/payo↵ function ui (s) =
ui (s1 , . . . , sI ), that determines player i?s payo↵ at every possible play of the game.
The payo↵s are usually determined by the rules of the game or economic phenom-
enon you are studying.
Examples:
(a) (Rock Paper Scissors): Recall there are two players i = 1, 2 and each has three
actions Si = {R, P, S} and the payo↵s can be represented in a table:
R P S
R 0,0 -1,1 1,-1
P 1,-1 0,0 -1,1
S -1,1 1,-1 0,0
(b) (Oligopoly where firms choose quantities): Recall that the firms’ names are
i = 1, 2, . . . , n. The firms? actions are outputs qi > 0. The firms’ payo↵s are
their profits, which depends on the demand. Price will depend on total output
so we write P (q1 + q2 + .. + qn ). (For example P = 50 q1 q2 ... qn .) Profit
also depends on a firm’s costs, we assume these only depend on their output
c(qi ). Hence
Profit of Firm i = Revenue Costs
= Output of i ⇥ Price Costs of i
= qi P (q1 + q2 + .. + qn ) c(qi )
This completes our initial description of a game in strategic form. We now want to allow
for the possibility that players act randomly. We don’t necessarily think that players are
genuinely randomising. But it may be a very good description of what the players think
the others are doing. You may know exactly how you’re going to play Paper-Scissors-Rock,
but I don’t. From my point of view your action looks random. We call random actions
Mixed 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 probability distributions on their pure
actions. We write the mixed strategy of player i as i . We will write the profile of mixed
strategies for all players as = ( 1 , . . . , I ).
Example 1: In paper, scissors rock Si = {R, P, S} and i = (p, q, 1 p q) where p is the
probability R is played, q is the probability P is played and 1 p q is the probability S
is played.
,ooh
i
a cot choosing paper I pto.qtl il.li p g
1: GAMES IN STRATEGIC FORM 3
Example 2: When firms choose quantities Si = [0, 1) is the set of possible quantities. A
random choice of a quantity can be represented by a probability distribution over the set of
positive numbers. One way of describing such a distribution is to write down its cumulative
distribution function (cdf) F (x) := P r(Firm’s outputs less than or equal to x).
Payo↵s from Mixed Actions: A player’s payo↵ when mixed actions are played is an expec-
tation (or average) taken over all the payo↵s they may get multiplied by the probability
they get them. This expectation is taken assuming the players randomise independently.
Examples of payo↵s from mixed actions (Rock Paper Scissors): Suppose in this game you
are the column player and you believe the row player will play action R with probability
p, P with probability q and S with probability 1 p q. You are interested in your payo↵
from action P. Below we have written out the payo↵s and emphasised the relevant numbers
for the column player.
R P S
p 0,0 -1,1 1,-1
q 1,-1 0,0 -1,1
1 p q -1,1 1,-1 0,0
If she plays P she expects to get 1 with probability p, 0 with probability q and 1 with
probability 1 p q. Thus on average she expects to get
p ⇥ 1 + q ⇥ 0 + (1 p q) ⇥ ( 1) = 2p + q 1.
If we repeat these calculations for each column we can work out what each action will give
her in expectation
R P S
p 0,0 -1,1 1,-1
q 1,-1 0,0 -1,1
1 p q -1,1 1,-1 0,0
1 p 2q 2p + q 1 q p
2. Dominance
2.1. Strict Dominance.
Definition 1. A mixed strategy i strictly dominates the pure action s0i for player i, if and
only if, player i’s payo↵ when she plays i and the other players play actions s i is strictly
higher than her payo↵ from s0i against s i for any actions s i the others may play:
ui ( i , s i ) > ui (s0i , s i ), 8s i .
Consider the following game (we only put in the row player’s payo↵s as that is all that
matters for now).
L R
T 3 0
M 0 3
B 1 1
, 4 MARTIN CRIPPS
And consider two strategies for the row player: play the first two rows with equal probability
i = (1/2, 1/2, 0), or play the bottom row si = B. We can write the expected payo↵s to
these strategies in the following way.
L R L R
1
2 3 0 0 3 0
1
2 0 3 0 0 3
0 1 1 1 1 1
3 3 1 1
2 2
From this you can see that if the column player plays L the strategy i gives the row player
the payo↵ of 32 but the strategy si gives the row player the payo↵ 1. And, if the column
player plays R strategy i also gives the row player the payo↵ of 32 but the strategy si gives
the row player the payo↵ 1. Thus the strategy i always does better than the strategy
si . To describe this we say si is strictly dominated and we would never expect a rational
player to play this.
However, eliminating strictly dominated actions can allow us to make strong predictions
about what actions the players will use. Consider the following game. . .
L R
U (8,10) (-100,9)
D (7,6) (6,5)
First observe that R is strictly dominated by L for the column player. So a rational column
player will never play R. If the row player knows this then they should play U getting
8 rather than 7. So we predict (U, L) as the outcome of this game. But, some types
of players may be very worried about the 100. If you had some doubts about column
player’s rationality would you be willing to play U ?
2.2. Weak Dominance. The notion of weak domination does not require players to
strictly prefer one strategy to another, it is enough to weakly prefer one to the other.
Definition 2. A mixed strategy i weakly dominates the pure action si 2 Si for player i,
if and only if, playing i is at least as good as playing si whatever the other players do:
ui ( i , s i ) ui (si , s i ), 8s i .
L R
T (1,1) (0,0)
M (0,0) (0,0)
B (0,0) (0,0)
T weakly dominates M for the row player, because T is better than M if column plays L
and T is no worse than M is column plays R.
Payoffs
Purestrategies
1: GAMES IN STRATEGIC FORM
MARTIN CRIPPS
The purpose of this topic is to teach you what a game in strategic form is and how
economists analyze them.
1. Games in Strategic Form: Description and Examples
1.1. Defining a Game in Strategic Form. A game in strategic form consists of several
elements. Here we are quite precise about what must be included in the description. There
must be the following three things:
(1) A list of players; i = 1, 2, . . . , I. This list can be finite or infinite. Here the names
of the players are just numbers but they can be any entity you are interested in
studying.
Examples:
(a) In Rock-Paper-Scissors there are two players i =Fred, Daisy.
(b) In an oligopoly with 5 firms there are n = 5 players i =Ford, Honda, Toyota,
VW, Fiat, GM.
(c) In bargaining there is a buyer and a seller, so the list of players is i =buyer,
seller.
(2) A description of all the actions each player can take. The set/list of possible
actions for player i is usually called their pure strategies by game theorists. We will
represent this by the set Si and we write a typical action as si 2 Si .
Examples:
(a) In Rock-Paper-Scissors, a player has three pure strategies: so
si 2 {Rock, Paper, Scissors} = Si , for i = 1, 2.
(b) In an oligopoly, each firm, i, will choose a non-negative quantity of output to
produce, 0 qi the set Si is the set of all positive output quantities.
(c) In bargaining, the buyer proposes a price 0 p 1 and the seller makes a
proposal too so the strategy sets are just intervals of prices: Sbuyer = {0
p 1}, Sseller = {0 p 1}.
When you want to write all the actions that were taken by all of the players
you will write a list or vector s = (s1 , . . . , sI ). 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 = (s1 , . . . , sI ) 2 S.
Examples:
1
,2 MARTIN CRIPPS
(a) In Rock-Paper-Scissors: an example of a profile is s = (Scissors, Paper) here
player 1 does S and 2 does P . The set of all possible profiles S has 9 elements.
(b) In oligopoly: a profile is a list of quantities for every firm each firm, i. That is
a list (q1 , q2 , . . . , qn ). The set of all possible profiles is Rn+ .
(c) Bargaining: a profile is two prices (pb , ps ) and the set of all possible profiles is
[0, 1]2 .
(3) The final element is the payo↵s (or utility or profit) the players get from their
actions or pure strategies. We can write this as a utility/payo↵ function ui (s) =
ui (s1 , . . . , sI ), that determines player i?s payo↵ at every possible play of the game.
The payo↵s are usually determined by the rules of the game or economic phenom-
enon you are studying.
Examples:
(a) (Rock Paper Scissors): Recall there are two players i = 1, 2 and each has three
actions Si = {R, P, S} and the payo↵s can be represented in a table:
R P S
R 0,0 -1,1 1,-1
P 1,-1 0,0 -1,1
S -1,1 1,-1 0,0
(b) (Oligopoly where firms choose quantities): Recall that the firms’ names are
i = 1, 2, . . . , n. The firms? actions are outputs qi > 0. The firms’ payo↵s are
their profits, which depends on the demand. Price will depend on total output
so we write P (q1 + q2 + .. + qn ). (For example P = 50 q1 q2 ... qn .) Profit
also depends on a firm’s costs, we assume these only depend on their output
c(qi ). Hence
Profit of Firm i = Revenue Costs
= Output of i ⇥ Price Costs of i
= qi P (q1 + q2 + .. + qn ) c(qi )
This completes our initial description of a game in strategic form. We now want to allow
for the possibility that players act randomly. We don’t necessarily think that players are
genuinely randomising. But it may be a very good description of what the players think
the others are doing. You may know exactly how you’re going to play Paper-Scissors-Rock,
but I don’t. From my point of view your action looks random. We call random actions
Mixed 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 probability distributions on their pure
actions. We write the mixed strategy of player i as i . We will write the profile of mixed
strategies for all players as = ( 1 , . . . , I ).
Example 1: In paper, scissors rock Si = {R, P, S} and i = (p, q, 1 p q) where p is the
probability R is played, q is the probability P is played and 1 p q is the probability S
is played.
,ooh
i
a cot choosing paper I pto.qtl il.li p g
1: GAMES IN STRATEGIC FORM 3
Example 2: When firms choose quantities Si = [0, 1) is the set of possible quantities. A
random choice of a quantity can be represented by a probability distribution over the set of
positive numbers. One way of describing such a distribution is to write down its cumulative
distribution function (cdf) F (x) := P r(Firm’s outputs less than or equal to x).
Payo↵s from Mixed Actions: A player’s payo↵ when mixed actions are played is an expec-
tation (or average) taken over all the payo↵s they may get multiplied by the probability
they get them. This expectation is taken assuming the players randomise independently.
Examples of payo↵s from mixed actions (Rock Paper Scissors): Suppose in this game you
are the column player and you believe the row player will play action R with probability
p, P with probability q and S with probability 1 p q. You are interested in your payo↵
from action P. Below we have written out the payo↵s and emphasised the relevant numbers
for the column player.
R P S
p 0,0 -1,1 1,-1
q 1,-1 0,0 -1,1
1 p q -1,1 1,-1 0,0
If she plays P she expects to get 1 with probability p, 0 with probability q and 1 with
probability 1 p q. Thus on average she expects to get
p ⇥ 1 + q ⇥ 0 + (1 p q) ⇥ ( 1) = 2p + q 1.
If we repeat these calculations for each column we can work out what each action will give
her in expectation
R P S
p 0,0 -1,1 1,-1
q 1,-1 0,0 -1,1
1 p q -1,1 1,-1 0,0
1 p 2q 2p + q 1 q p
2. Dominance
2.1. Strict Dominance.
Definition 1. A mixed strategy i strictly dominates the pure action s0i for player i, if and
only if, player i’s payo↵ when she plays i and the other players play actions s i is strictly
higher than her payo↵ from s0i against s i for any actions s i the others may play:
ui ( i , s i ) > ui (s0i , s i ), 8s i .
Consider the following game (we only put in the row player’s payo↵s as that is all that
matters for now).
L R
T 3 0
M 0 3
B 1 1
, 4 MARTIN CRIPPS
And consider two strategies for the row player: play the first two rows with equal probability
i = (1/2, 1/2, 0), or play the bottom row si = B. We can write the expected payo↵s to
these strategies in the following way.
L R L R
1
2 3 0 0 3 0
1
2 0 3 0 0 3
0 1 1 1 1 1
3 3 1 1
2 2
From this you can see that if the column player plays L the strategy i gives the row player
the payo↵ of 32 but the strategy si gives the row player the payo↵ 1. And, if the column
player plays R strategy i also gives the row player the payo↵ of 32 but the strategy si gives
the row player the payo↵ 1. Thus the strategy i always does better than the strategy
si . To describe this we say si is strictly dominated and we would never expect a rational
player to play this.
However, eliminating strictly dominated actions can allow us to make strong predictions
about what actions the players will use. Consider the following game. . .
L R
U (8,10) (-100,9)
D (7,6) (6,5)
First observe that R is strictly dominated by L for the column player. So a rational column
player will never play R. If the row player knows this then they should play U getting
8 rather than 7. So we predict (U, L) as the outcome of this game. But, some types
of players may be very worried about the 100. If you had some doubts about column
player’s rationality would you be willing to play U ?
2.2. Weak Dominance. The notion of weak domination does not require players to
strictly prefer one strategy to another, it is enough to weakly prefer one to the other.
Definition 2. A mixed strategy i weakly dominates the pure action si 2 Si for player i,
if and only if, playing i is at least as good as playing si whatever the other players do:
ui ( i , s i ) ui (si , s i ), 8s i .
L R
T (1,1) (0,0)
M (0,0) (0,0)
B (0,0) (0,0)
T weakly dominates M for the row player, because T is better than M if column plays L
and T is no worse than M is column plays R.
