LECTURE 1: GAMES IN STRATEGIC FORM
A. Games in Strategic Form: Description and Examples
- A list of players, i = 1,2,…,I
- I can be finite or infinite.
- Examples:
- In Rock-Paper-Scissiors there are two players → i = 1,2.
- In an oligopoly with n firms there are n players.
- In bargaining there is a buyer and a seller, so the list of players is i = buyer,
seller.
- Actions or pure strategies for each player; Si can also be finite or infinite.
- A typical action is si ∈ Si.
- Examples of action sets:
- In Rock-Paper-Scissors, a player has strategies: so si ∈ {Rock, Paper,
Scissors} = Si, for i = 1,2.
- In an oligopoly, each firm, i, will choose a quantity, 0 ≤ qi → the set Si is the
set of all positive quantities.
- In bargaining, the buyer proposes a price 0 ≤ p ≤ 1 and the seller makes a
proposal too.
Sbuyer = {0 ≤ p ≤ 1}
Sseller = {0 ≤ p ≤ 1}
- An action profile is s = (s1 ,…, sI) ∈ S is a list of actions for all the players in the
game.
- Examples of strategy profiles:
- Rock-Paper Scissors: an example of a profile is s = (Scissors, Paper) →
player 1 does S and 2 does P.
- Oligopoly: a profile is a list of quantities for every firm → each firm, i, will
choose a quantity, (q1 , q2 ,…, qn ).
- Bargaining: a profile is two prices (pb, ps ).
- The payoffs (or utility or profit) the players get from their actions or pure strategies:
ui (s) = ui (s1 ,…, sI )
- This is a function that tells you what each player’s payoff is for each possible action
profile —> usually determined by the rules of the game.
Example 1: Rock Paper Scissors
- Two players → i = 1,2
- Each with three actions Si = {R,P,S}
- Payoffs:
,Example 2: Oligopoly where firms choose quantities
- Players (i.e. firms) → i = 1,2, …, n
- Each chooses and output → 0 ≤ qi
- Payoffs:
- Profits of Firm i = Revenue - Costs
- Suppose that each firm’s costs only depends on their output Costs = c(qi)
- Revenue = Price x Quantity = qiP(q1+q2+…+qn)
- Profits of Firm i = qiP(q1+q2+…+qn) - c(qi)
- Suppose that P = 50 - qi - q2 … - qn and that c(qi) = 2qi:
Payoff/ profit = qi [50 - q1 - q2 - … - qn] - 2qi
Is a perfectly competitive market a game?
- No: In a perfectly competitive market, agents choose how much to demand or supply.
They choose quantities. There is no way to explain how the price is determined.
- Random actions (mixed strategies) → we want to allow the players to randomise
in their choice of an action.
- It is not necessarily the case that players actually behave randomly.
- What is important is what player A thinks player B is going to do. As player A
may not know what player B is going to do, B’s actions should be treated as
being random by player A.
- A player’s set of mixed strategies is the set of all probabilities on their pure actions.
- The mixed strategy of player i is written as σi.
- The profile of mixed strategies of all players is written as σ = (σ1, …, σI)
- Example 1:
- Si = {R,P,S} and σi = (p, q, 1-p-q)
- Example 2:
- Si = [0, ∞] a set of possible quantities mixed strategy is a cumulative
distribution function (cdf):
F(x) = Pr(quantity is less than or equal to x)
- Payoffs from mixed actions:
- A player’s playoff is an expectation taken over their random action and all the
other random actions of all their opponents.
- This expectation is taken assuming the players randomise independently.
,
, B. Dominance Arguments: Iterated strict and weak dominance
- s-i → describes a list of actions for all the players who are not player i.
s-1 = (s2,s3,…,sI)
s-2 = (s1,s3,…,sI)
Strict domination
- A mixed strategy σi strictly dominates the pure action s’i for player i, if and only if:
Player i’s payoff when she plays σi and all the other players play the action s-i is
slightly higher than her payoffs from s’i against s-i for any actions s-i the others may
play.
ui (σi , s-i) > ui (s’i , s-i ), for all s-i
Example:
(in this example we focus only on the row player’s playoffs)
Weak dominance
A. Games in Strategic Form: Description and Examples
- A list of players, i = 1,2,…,I
- I can be finite or infinite.
- Examples:
- In Rock-Paper-Scissiors there are two players → i = 1,2.
- In an oligopoly with n firms there are n players.
- In bargaining there is a buyer and a seller, so the list of players is i = buyer,
seller.
- Actions or pure strategies for each player; Si can also be finite or infinite.
- A typical action is si ∈ Si.
- Examples of action sets:
- In Rock-Paper-Scissors, a player has strategies: so si ∈ {Rock, Paper,
Scissors} = Si, for i = 1,2.
- In an oligopoly, each firm, i, will choose a quantity, 0 ≤ qi → the set Si is the
set of all positive quantities.
- In bargaining, the buyer proposes a price 0 ≤ p ≤ 1 and the seller makes a
proposal too.
Sbuyer = {0 ≤ p ≤ 1}
Sseller = {0 ≤ p ≤ 1}
- An action profile is s = (s1 ,…, sI) ∈ S is a list of actions for all the players in the
game.
- Examples of strategy profiles:
- Rock-Paper Scissors: an example of a profile is s = (Scissors, Paper) →
player 1 does S and 2 does P.
- Oligopoly: a profile is a list of quantities for every firm → each firm, i, will
choose a quantity, (q1 , q2 ,…, qn ).
- Bargaining: a profile is two prices (pb, ps ).
- The payoffs (or utility or profit) the players get from their actions or pure strategies:
ui (s) = ui (s1 ,…, sI )
- This is a function that tells you what each player’s payoff is for each possible action
profile —> usually determined by the rules of the game.
Example 1: Rock Paper Scissors
- Two players → i = 1,2
- Each with three actions Si = {R,P,S}
- Payoffs:
,Example 2: Oligopoly where firms choose quantities
- Players (i.e. firms) → i = 1,2, …, n
- Each chooses and output → 0 ≤ qi
- Payoffs:
- Profits of Firm i = Revenue - Costs
- Suppose that each firm’s costs only depends on their output Costs = c(qi)
- Revenue = Price x Quantity = qiP(q1+q2+…+qn)
- Profits of Firm i = qiP(q1+q2+…+qn) - c(qi)
- Suppose that P = 50 - qi - q2 … - qn and that c(qi) = 2qi:
Payoff/ profit = qi [50 - q1 - q2 - … - qn] - 2qi
Is a perfectly competitive market a game?
- No: In a perfectly competitive market, agents choose how much to demand or supply.
They choose quantities. There is no way to explain how the price is determined.
- Random actions (mixed strategies) → we want to allow the players to randomise
in their choice of an action.
- It is not necessarily the case that players actually behave randomly.
- What is important is what player A thinks player B is going to do. As player A
may not know what player B is going to do, B’s actions should be treated as
being random by player A.
- A player’s set of mixed strategies is the set of all probabilities on their pure actions.
- The mixed strategy of player i is written as σi.
- The profile of mixed strategies of all players is written as σ = (σ1, …, σI)
- Example 1:
- Si = {R,P,S} and σi = (p, q, 1-p-q)
- Example 2:
- Si = [0, ∞] a set of possible quantities mixed strategy is a cumulative
distribution function (cdf):
F(x) = Pr(quantity is less than or equal to x)
- Payoffs from mixed actions:
- A player’s playoff is an expectation taken over their random action and all the
other random actions of all their opponents.
- This expectation is taken assuming the players randomise independently.
,
, B. Dominance Arguments: Iterated strict and weak dominance
- s-i → describes a list of actions for all the players who are not player i.
s-1 = (s2,s3,…,sI)
s-2 = (s1,s3,…,sI)
Strict domination
- A mixed strategy σi strictly dominates the pure action s’i for player i, if and only if:
Player i’s payoff when she plays σi and all the other players play the action s-i is
slightly higher than her payoffs from s’i against s-i for any actions s-i the others may
play.
ui (σi , s-i) > ui (s’i , s-i ), for all s-i
Example:
(in this example we focus only on the row player’s playoffs)
Weak dominance
