# 1. Forms and Representations

```Games &amp; Strategy
1. Introduction and Representations of a
Game
(Chapters 2, 3, 4, and 5)
1
Overview
• There are various forms of a game
▪ strategic games (also referred to as normal-form games)
▪ extensive-form games
▪ repeated games
▪ Bayesian games
2
Overview
• An example of a strategic game: Prisoners’ dilemma
Suspect 2
Suspect 1
Quiet
Confess
Quiet
2, 2
0, 3
Confess
3, 0
1, 1
3
Overview
• An example of an extensive-form game
4
Overview
• Prisoners’ dilemma can also be played as an extensive-form
game
▪ But now it is a different game
5
Overview
• An example of a repeated game
▪ Suppose that the Prisoners’ dilemma game is repeated many times
▪ Is there any incentive for players to cooperate?
6
Overview
• An example of an incomplete-information game
7
Overview
• A strategic game consists of
▪ a set of players
▪ for each player, a set of actions
▪ for each player, preferences over the set of action profiles
• See, again, the prisoners’ dilemma
Suspect 2
Suspect 1
Quiet
Confess
Quiet
2, 2
0, 3
Confess
3, 0
1, 1
8
Overview
• In the prisoners’ dilemma game
▪ the set of players: {Suspect 1, Suspect 2}
▪ for Suspect 1, the set of actions: {Quiet, Confess}
▪ for Suspect 2, the set of actions: {Quiet, Confess}
▪ Suspect 1’s preference orderings over action profiles:
(confess, quiet) &gt; (quiet, quiet) &gt; (confess, confess) &gt; (quiet, confess)
▪ Suspect 2’s preference orderings over action profiles:
(quiet, confess) &gt; (quiet, quiet) &gt; (confess, confess) &gt; (confess, quiet)
9
Extensive-form representation
• We represent an extensive-form game as a game tree
▪ Nodes represent places where something happens in the game
(usually, decisions and termination)
▪ Branches indicate the various actions that players can choose
▪ Such graphical representation is called an extensive-form
representation
10
Extensive-form representation
• Consider the following extensive-form game
11
Extensive-form representation
• In this game,
▪ Players are K (Katzenberg) and E (Eisner)
▪ Each branch indicates their actions
▪ Node a is the initial node
▪ Nodes f, g, h, l, m, and n are terminal nodes (outcomes of the game)
▪ Each player has preference orderings over terminal nodes
▪ Their preferences are represented by payoffs or utilities
12
Extensive-form representation
• We use an information set to specify the players’ information
at each decision node
▪ Dashed line indicates that the player cannot distinguish between the
nodes connected by the dashed line
▪ For example, Katzenberg doesn’t know whether he is at c or d
▪ In other words, he doesn’t know whether Eisner decides to produce
“A Bug’s Life” or not.
13
Extensive-form representation
• If several nodes are in the same information set, it implies that the
player cannot distinguish between the nodes in the information
set
▪ For example, when Katzenberg decides whether to leave or stay, he knows
he is at a, so information set is {a}
▪ At Eisner’s turn, he knows he is at b (he knows that Katzenberg has left),
so information set is {b}
▪ But Katzenberg does not know whether Eisner decided to produce or not,
in other words, Katzenberg does not know whether he is at c or d, so
information set is {c, d}
▪ After his own decision to produce “Antz”, Katzenberg knows he is at e, so
information set is {e}
14
Some terminology
• Consider the following price-competition game
▪ We use difference labels (H and H’, L and L’) for player 2’s decision
to avoid ambiguity
15
Extensive-form representation
• Consider the following price-competition game
▪ Although this game looks a game in which the decisions are made
sequentially, it is equivalent to a simultaneous-move game
16
Strategy
• A strategy is a complete contingent plan for a player
▪ A “complete contingent” plan means a full specification of a player’s
behavior
▪ It describes the actions that the player would take at each of his
possible decision points
▪ It describes what the player will do at each of his information sets
17
Strategy
• Consider again the following game
18
Strategy
• Player 1’s strategy:
▪ play H
▪ play L
• Player 2’s strategy:
▪ play H if player 1 plays H and play H’ if player 1 plays L
▪ play H if player 1 plays H and play L’ if player 1 plays L
▪ play L if player 1 plays H and play H’ if player 1 plays L
▪ play L if player 1 plays H and play L’ if player 1 plays L
19
Strategy
• Compare with this variant
20
Strategy
• Player 1’s strategy is same as before
▪ play H
▪ play L
• Player 2’s strategy:
▪ play H
▪ play L
21
Strategy
• Strategy space (strategy set) of player i (𝑆𝑖 ) is a set
comprising each of the possible strategies of player i
▪ In the previous game,
𝑆1 ={H, L}
𝑆2 ={HH’, HL’, LH’, LL’)
▪ We let 𝑆𝑖 denote a strategy set of player i and 𝑠𝑖 a single strategy
for example, 𝑠1 =H, 𝑠2 =HL’
22
Strategy
• A strategy profile is a vector of strategies, one for each player
▪ In the previous case, there are 8 possible strategy profiles
((H,HH’),(H,HL’),(H,LH’),(H,LL’),(L,HH’),(L,HL’),(L,LH’),(L,LL’))
▪ A typical strategy profile is a vector s=(𝑠1 , 𝑠2 ,…,𝑠𝑛 ) where n is the number of
players (i=1,2,….,n)
▪ Let S denote the set of strategy profiles (S=𝑆1 &times; 𝑆2 &times; ⋯ &times; 𝑆𝑛 )
▪ In the above example,
S= 𝑆1 &times; 𝑆2 ={(H,HH’),(H,HL’),(H,LH’),(H,LL’),(L,HH’),(L,HL’),(L,LH’),(L,LL’)}
23
Strategy
• We usually use the term –i to refer everyone except player i
▪ 𝑠−𝑖 is a strategy profile for everyone except player i
𝑠−𝑖 = (𝑠1 , 𝑠2 , … , 𝑠𝑖−1 , 𝑠𝑖+1 , … , 𝑠𝑛 )
▪ We often simply write 𝑠 = (𝑠𝑖 , 𝑠−𝑖 )
24
Strategy
• In games (a) and (b) (separately), find the strategy sets of
each player
25
Normal form
• For each player i, we can define a function 𝑢𝑖 : 𝑆 → 𝑹 (𝑹 is the
set of real numbers)
▪ In other words, each strategy profile is associated with a real number
(payoff or utility) for each player
▪ So for each strategy profile 𝑠 ∈ 𝑆, 𝑢𝑖 𝑠 is player i’s payoff
▪ 𝑢𝑖 is called player i’s payoff function
26
Normal form
• Again in the following game
▪ S= 𝑆1 &times; 𝑆2 ={(H,HH’),(H,HL’),(H,LH’),(H,LL’),(L,HH’),(L,HL’),(L,LH’),(L,LL’)}
▪ for example, 𝑢1 (𝐿, 𝐻𝐻′) = 2, 𝑢2 (𝐻, 𝐿𝐻′) = 2
27
Normal form
• In the following game,
▪ 𝑆1 ={OA, OB, IA, IB}, 𝑆2 ={O, I}
▪ S= 𝑆1 &times; 𝑆2 ={(OA,O),(OA,I),(OB,O),(OB,I),(IA,O),(IA,I),(IB,O),(IB,I)}
▪ For example, 𝑢1 (OA,O)=2, 𝑢1 (IA,I)=4, 𝑢2 (IA,O)=3
28
Normal form
• The strategy sets and payoff functions fully describe a
strategic situation (without reference to an extensive form)
• A game in normal form (or strategic form) consists of
▪ a set of players: {1, 2, …, n}
▪ strategy space for the players: 𝑆1 , 𝑆2 , … , 𝑆𝑛
▪ payoff functions for the players : 𝑢1 , 𝑢2 , … , 𝑢𝑛
29
Normal form
• Two-player normal-form games with finite strategy spaces
can be described by matrices (matrix game)
• The previous extensive-form game can be represented as a
normal-form game
2
1
I
O
OA
2, 2
2, 2
OB
2, 2
2, 2
IA
4, 2
1, 3
IB
3, 4
1, 3
30
Belief
• Belief is a player’s assessment about the strategies of the
others in the game
▪ A belief of player i is a probability distribution over the strategies of
the other players
▪ Denote such a probability distribution 𝜃−𝑖
▪ 𝜃−𝑖 ∈ ∆𝑆−𝑖 where ∆𝑆−𝑖 is the set of probability distributions over the
strategies of all the players except player i
31
Mixed Strategy
• A mixed strategy for a player is the act of selecting a strategy
according to a probability distribution
▪ We denote a generic mixed strategy of player i by 𝜎𝑖 ∈ ∆𝑆𝑖
▪ We call a regular strategy a pure strategy (i.e. chooses one action
with 100% probability)
32
Expected Value (Payoff)
• When player i has a belief about the strategies of other
players, we can calculate the expected value (or expected
payoff) of each action in her strategy set
▪ Assume that player i has a belief 𝜃−𝑖 about the strategies of the
others and plans to select 𝑠𝑖
▪ Then expected payoff is 𝑢𝑖 𝑠𝑖 , 𝜃−𝑖 = σ𝑠−𝑖 ∈𝑆−𝑖 𝜃−𝑖 (𝑠−𝑖 )𝑢𝑖 (𝑠𝑖 , 𝑠−𝑖 )
(don’t worry if you don’t understand this mathematical expression)
(examples to be shown in class)
33
Rationality
• Rationality means maximizing one’s expected payoff
▪ Each player acts to maximize his/her own expected payoff
• A player is thought to be rational if
▪ through some cognitive process, he/she forms a belief about the
strategies of the others
▪ given this belief, he/she selects a strategy to maximize his/her
expected payoff
▪ Nash equilibrium requires additional assumption that players’ beliefs