CAP 5507 – Module 2
Lecture 1 – Strategies and Normal Forms:
Strategy:
This is the most important concept in game theory (interactions
between two players and a game).
Strategies refer to a complete contingent plan for a player in the game.
o A complete contingent plan is a full specification of a player’s
behavior, describing the actions that the player would take at
each of their possible decision points.
o Recall: A decision node is an area where the player decides.
o Decisions are based on an information set, since they represent
places in the game at which players make decisions, and a
player’s strategy will describe what is done at the information
set.
Player 1 has one information set and player 2 has
two of them.
Player 2 can either take H or H’, depending on the
node where the player is at.
Player 2 might have the strategy of a cross involving
the cross of actions to take: HH’, HL’, LH’, LL’ (cross
the elements from a strategy to another one)
Player 1 only has two strategies: H, L.
When we count the number of individual sets, count
the number of letters for each individual player. Each
strategy must have a certain number of letters.
o The strategy tells the player what a player
does within the nodes.
More Formal Structure of a Strategy:
Given a game, we use Si to denote a strategy space, or a strategy
set for player set.
o S1={ H , L }, S2={ HH ' , HL ' , LH ' ,≪' }
Lowercase si ∈ Si refers to an element in player i’s strategy set.
o Example: si = L, s2=L H '
, A strategy profile is a vector of strategies from each player,
describing strategies of all players of the game.
o Strategy profile, s = ( s1 , s 2 , ... s n ¿ , we denote S to be the set of
strategy profiles: S = S1 x S2 x S3… x Sn
o An alternate notation is that if we are given a single player I,
we might have to consider the strategies chosen by other
players:
S−i =( s1 , s 2 , ... s i−1 , s i+1 , ... , sn ).
o Another useful notation in the separation of a strategy profile
s into player siand s−i : s = ( si , s−i )
Example: We have an extended form game. When we
see a game, identify the players, decision nodes, and
information sets.
Consider the fact that we have two players, based on
their nodes.
Player 1 has 1 information set, and so does player 2,
because it has a dotted line, indicating that it cannot
see player 1’s A or P and does not know what it chose.
Player 2 knows that player 1 did O since there is no
dotted line there.
In this game, we model if a firm wants to be aggressive
(A), passive (P), or exit a market (O).
Firm 1 decides if they want to be A, P, or O, which
are all actions that player 1 can do at the node.
If firm 1 leaves, firm 2 has a monopoly (no
competition).
Recall that from extended forms, we have
terminating nodes (numbers / payouts) at the end.
If player 1 chooses O, they get a payout of 0, and
player 2 gets a payout of 4.
Firm 1 strategy: {A, P, O}, there is only 1 letter
because there is 1 information set.
o Firm 2 has one information set. It only
knows if firm 1 is out of the market, and it
does not know the competition level. It is
Lecture 1 – Strategies and Normal Forms:
Strategy:
This is the most important concept in game theory (interactions
between two players and a game).
Strategies refer to a complete contingent plan for a player in the game.
o A complete contingent plan is a full specification of a player’s
behavior, describing the actions that the player would take at
each of their possible decision points.
o Recall: A decision node is an area where the player decides.
o Decisions are based on an information set, since they represent
places in the game at which players make decisions, and a
player’s strategy will describe what is done at the information
set.
Player 1 has one information set and player 2 has
two of them.
Player 2 can either take H or H’, depending on the
node where the player is at.
Player 2 might have the strategy of a cross involving
the cross of actions to take: HH’, HL’, LH’, LL’ (cross
the elements from a strategy to another one)
Player 1 only has two strategies: H, L.
When we count the number of individual sets, count
the number of letters for each individual player. Each
strategy must have a certain number of letters.
o The strategy tells the player what a player
does within the nodes.
More Formal Structure of a Strategy:
Given a game, we use Si to denote a strategy space, or a strategy
set for player set.
o S1={ H , L }, S2={ HH ' , HL ' , LH ' ,≪' }
Lowercase si ∈ Si refers to an element in player i’s strategy set.
o Example: si = L, s2=L H '
, A strategy profile is a vector of strategies from each player,
describing strategies of all players of the game.
o Strategy profile, s = ( s1 , s 2 , ... s n ¿ , we denote S to be the set of
strategy profiles: S = S1 x S2 x S3… x Sn
o An alternate notation is that if we are given a single player I,
we might have to consider the strategies chosen by other
players:
S−i =( s1 , s 2 , ... s i−1 , s i+1 , ... , sn ).
o Another useful notation in the separation of a strategy profile
s into player siand s−i : s = ( si , s−i )
Example: We have an extended form game. When we
see a game, identify the players, decision nodes, and
information sets.
Consider the fact that we have two players, based on
their nodes.
Player 1 has 1 information set, and so does player 2,
because it has a dotted line, indicating that it cannot
see player 1’s A or P and does not know what it chose.
Player 2 knows that player 1 did O since there is no
dotted line there.
In this game, we model if a firm wants to be aggressive
(A), passive (P), or exit a market (O).
Firm 1 decides if they want to be A, P, or O, which
are all actions that player 1 can do at the node.
If firm 1 leaves, firm 2 has a monopoly (no
competition).
Recall that from extended forms, we have
terminating nodes (numbers / payouts) at the end.
If player 1 chooses O, they get a payout of 0, and
player 2 gets a payout of 4.
Firm 1 strategy: {A, P, O}, there is only 1 letter
because there is 1 information set.
o Firm 2 has one information set. It only
knows if firm 1 is out of the market, and it
does not know the competition level. It is
