1
Representation of Games
Kerschbamer: Commitment and Information in Games
Game-Theoretic Description of Interactive Decision Situations
This lecture deals with the process of translating an informal description of an interactive decision situation in a game-theoretic problem
There are two basic forms or types of formal models to describe interactive decision situations:
– normal-form (or strategic-form) representation
– extensive-form representation
Normal-Form Representation
Definition 1: The normal-form (or strategic form) representation of a (finite) game specifies 1. Players. A set of players (agents who play the game) N = {1,…, n} with typical element i ∈
N
2. Strategies. For each player i ∈ N a nonempty set of feasible strategies Siwith typical element si ∈ Si
3. Payoffs. For each player i ∈ N a payoff function ui: S ↦
ℝ
, where S = xi ∈N Si. Notation: s ∈ S = xi ∈N Si is called a “strategy profile” or a “strategy combination”.Note: The payoff function uiof player i specifies i’s payoff for each strategy profile in S.
A formal way to write down the normal-form of a game is GN = [N, {Si}i∈ N , {ui}i∈ N ].
For “simple” games a convenient way to summarize the normal-form information is the bi-matrix form (see next slides).
Example 1: “Rock-Paper-Scissors”
• there are two children who simultaneously choose one of three options
• the three options are rock, paper, scissors
• if the two choose the same option, the game is a draw; if one chooses rock and the second paper, the second wins (paper covers rock); if one chooses rock and the second scissors, the first wins (rock breaks scissors); and if one chooses paper and the second scissors, the second wins (scissor cuts paper)
normal-form representation, formal version:
Players. The players are the two children: N = {1,2}
Strategies. The strategy set for the two players are S1 = {R, P, S} and S2 = {R, P, S}
Payoffs. The payoffs of the two players are:
u2(R, R) = 0 u2(R, P) = 1 u (R, S) = -1 u1(R, R) = 0
u1(R, P) = -1 u (R, S) = 1
u2(s1,s2) u1(s1,s2)
Bi-Matrix Representation of “Rock-Paper-Scissors”
0, 0 1, -1
-1, 1 Scissors
-1, 1 0, 0
1, -1 Paper
1, -1 -1, 1
0, 0 Rock
Scissors Paper
Rock
child 2 child 1
payoff child 1
payoff child 2 strategies
of
Example 2: “Matching Pennies”
• there are two players, denoted A and B
• each player simultaneously puts a penny (an Euro) down, either heads up (H) or tails up (T)
• if the two pennies match (either both H or both T), player A pays 1 Dollar (1 Euro) to player B; otherwise player B pays 1 Dollar to player A.
normal-form representation, formal version:
Players. N = {………}
Strategies. SA = {………}; SB = {………}
Payoffs. uA(sA, sB) = ………, uB(sA, sB) = … ………
normal-from representation, bi-matrix version:
T
sB
H
sA
Normal-Form Representation of Games with Continuous Strategy Spaces Example 3: Cournot - Duopoly
• homogeneous products market
• two firms simultaneously choose output quantities s1 and s2 smaller or equal to 80
• total output: x = s1 + s2
• inverse demand: P(x) = max{80 - x; 0}
• cost functions: C1(s1) = 8s1; C2(s2) = 2s2
normal-form representation (adapt Definition 1):
Players. N = {1, 2}
Strategies. S1 = [0, 80]; S2 = [0, 80]
Payoffs. - for s1 + s2 ≤ 80: u1(s1 , s2) = (72 – s1 – s2)s1 ; u2(s1 , s2) = (78 – s1 – s2)s2 - for s1 + s2 > 80: u1(s1 , s2) = – 8s1 ; u2(s1 , s2) = – 2s2
Normal-Form Representation of Games with Continuous Strategy Spaces Example 4: “Nash Demand Game”
• two individuals, 1 and 2, argue over the division of a (perfectly divisible) Dollar
• they simultaneously make irrevocable demands, s1 and s2, smaller or equal to the Dollar
• if the two demands sum to no more than the Dollar, then both player get their demand, otherwise, neither player receives any money
• both players are interested in their own monetary payoff only normal-form representation (adapt Definition 1):
Players. N = {………}
Strategies. S1 = …………; S2 = ………
Payoffs. u1(s1 , s2) = ……… ; u2(s1 , s2) = ………
Extensive-Form Representation
The normal-form representation is a very condensed representation of a game. It contains no info. on the timing of moves, on the actions available at each opportunity to move, etc. It seems, that only simultaneous move games can be represented in normal form. This is not true. More on this below…
Definition 2a (informal version): The extensive-form representation of a game specifies
• the players (agents) in the game,
• when each player has the move,
• what each player can do at each of her opportunities to move,
• what each player knows at each of her opportunities to move,
• in games with chance moves: the probabilities assigned to each feasible “move”,
• what the outcome is as a function of the actions taken by the players (inclusive the chance player
“nature”)
• the payoffs of the players (exclusive the chance player) from each possible outcome
For “simple” games a convenient way to summarize the extensive-form information is the game tree (see next slides).
We begin by informally introducing the elements of the extensive-form representation through a series of examples.
Example 5: Game Tree of a Game with Perfect Information
• game starts at an initial decision node
• at the initial decision node, player 1 makes her move
• her choice is between the two actions left (l) and right (r)
• each of the two possible actions of player 1 is represented by a branch from the initial decision node
• at the end of each branch is another decision node
• now player 2 can choose between two actions, left (L) and right (R)
• if player 1 has chosen l and player 2 has chosen L, we reach the end of the game, represented by a terminal node
• each terminal node lists the players’ payoffs
Example 6: Game Tree of a Game with Imperfect Information
• in example 5, when it is a player’s turn to move, she is able to observe all her rival’s previous moves
• such games are called games of perfect
information (we give a more precise definition below)
• the concept of an information set allows us to accommodate the possibility that this is not so
• the elements of an information set are a subset of a particular player’s decision nodes
• the interpretation is that when play has reached one of the decision nodes in the info set and it is that player’s turn to move, she does not know which of these nodes she is actually at
• note that player 2 has the same two possible actions at each of the two nodes in her info set
• this must be the case if player 2 is unable to distinguish the two nodes
Definition 3. An information set for a player is a collection of decision nodes satisfying: (i) the player has the move at every node in the information set; and (ii) when the play of the game reaches a node in the information set, the player with the move does not know which node in the set has been reached
Note: The use of info sets also allows us to capture play that is simultaneous in the game tree. Try Example 1 and Example 2.
Example 7 (Russian Roulette): Game Tree of a Game with Chance Moves
• in previous examples the outcome of the game has been a deterministic function of the
players’ choices
• in many games there is an element of chance
• this too, can be captured in the game tree by including random moves of nature (player 0)
• here is the story: two officers who have been competing for the affections of a Muscovite lady for a long time decide to settle the matter with the following game:
• a bullet is loaded at random into one of the chambers of a six-shooter
• the two players then alternate in taking turns
• when it is his turn, a player may chicken out (N) or point the gun at his own head and pull the trigger (A).
• chickening out or death disqualifies a player from further pursuit of the lady
• each player prefers being left with the lady undisturbed to chicken out and each prefers
Example 8: Random Moves and Information Sets
• two players flip a coin yielding H (for “heads up”) and T (for “tails up”) each with probability ½
• player 1 puts her penny down, either H or T
• player 2 puts his penny down, either H or T
• if the pennies of the two players match, both get 2 Euros
• if the pennies of the two players don’t match, the player whose penny matches nature’s move gets 5 Euro, the other player gets nothing
Draw the game tree for the following variants of the game:
Variant A: player 1 observes the outcome of the coin flip, player 2 observes neither the outcome of the coin flip nor player 1’s move
Variant B: player 1 observes the outcome of the coin flip, player 2 does not observe the outcome of the coin flip but he observes player 1’s move
Variant C: both players observe the outcome of the coin flip, player 2 does not observe player 1’s move
Variant D: both players observe the outcome of the coin flip, player 2 observes player 1’s move Variant E: no player observes the outcome of the coin flip, player 2 doesn’t observe player 1’s move Variant …
Extensive-Form Representation: Formal Definition
Definition 2b (formal version). The extensive-form representation of a game specifies 1. Players. A set of players N with typical element i.
2. Histories. A set of histories H with typical element h. Each h is a sequence of actions by
individual players. ∅ ∈ H is the start of the game. If h ∈ H, but there is no (h, a) ∈ H where a is an action for some player, then h is “terminal”. Denote the set of terminal histories as T ⊂ H 3. Player Function. A function P : H \ T ↦ N ∪ {0}, assigning a player or “nature” (formally
player 0) to each non-terminal history.
4. Nature. For each h ∈ H such that P(h) = 0, f(a | h) is the probability that (h, a) ∈ H occurs.
5. Information. For each player i ∈ N an information partition Iiof {h ∈ H : P(h) = i}. (h, a) ∈ H ⇔ (h′, a) ∈ H for all histories h, h′ ∈ H in the information set Ii ∈ Ii
6. Payoffs. vNM payoffs for each i ∈ N are defined over terminal histories, ui : T ↦
ℝ
Note: (h, a) is the history (of length t + 1) which consists of h (of length t), followed by a.
A formal way to write down the extensive-form of a game is
Applying the Formal Definition
Example 5:
Players. The set of players is N = {1, 2}. Nature does not move.
Histories. The set of histories is H = {(∅), (l), (r), (l, L), (l, R), (r, L), (r, R), (r, R, L), (r, R, R), (r, R, L, l), (r, R, L, r)}
The set of terminal histories is T = {(l, L), (l, R), (r, L), (r, R, R), (r, R, L, l), (r, R, L, r)}
Player Function. P(∅) = 1, P(l) = 2, P(r) = 2, P(r, R) = 2, P(r, R, L) = 1
Information. The information partition of player 1 is I1= {∅, {(r, R, L)}}, the information partition of player 2 is I2= {{(l)}, {(r)}, {(r, R)}}
Payoffs. Defined over T and shown in the tree, e.g. u1(l, L) = 5 and u2(l, L) = 0 Example 6:
Players. The set of players is N = {1, 2}. Nature does not move.
Histories. The set of histories is H = {………}
The set of terminal histories is T = {………}
Player Function. P(∅) = ………
Information. The information partition of player 1 is I1= ………
The information partition of player 2 is I2= …………..
Payoffs. ………
Applying the Formal Definition (Cont.)
Example 7:
Players. The set of players is N = {………}
Histories. The set of histories is H = {……… } The set of terminal histories is T = {……….}
Player Function. P(∅) = ………
Nature: f(1|∅) = …………
Information. The information partition of player 1 is I1= {………}
The information partition of player 2 is I2= {………}
Payoffs. ……….
Finiteness Assumptions in the Definition of an Extensive Game
Note: There are three types of finiteness assumptions hidden in Definition 2b
• Definition 2 implicitly assumes that players have a finite number of actions available at each decision node. Many economic applications (e.g. Cournot- and Bertrand-competition) violate this assumption. Allowing for an infinite set of actions requires that we allow for an infinite set of histories as well. Also, if nature has an infinite set of ‘moves’ the f function has to be adapted to allow for this.
• Definition 2 implicitly assumes that the game must end after a finite number of steps. Some economic situations (e.g. market interactions between firms) violate this assumptions. In games with an infinite sequence of moves there are no terminal nodes and no terminal
histories. Payoffs have to be defined over (infinite) sequences of moves (or infinite histories) instead of defining them over terminal nodes (or terminal histories).
• Definition 2 implicitly assumes a finite number of players who take actions in the game.
Some economic examples (e.g. overlapping generation models) violate this assumption.
Allowing for an infinite number of players requires some adaptations in the definition.
The formal definition of an extensive-form representation of a game can be extended to those infinite cases without much difficulty. We do not extend the definition here but we will adapt it when needed.
Strategies
• in normal-form games:
A strategy for a player in a normal-form game is simply one of the choices available to her in the game.
• in extensive-form games:
Definition 4a (informal version): A strategy for a player in an extensive-form game is a complete contingent plan, or decision rule, that specifies how the player will act in every possible distinguishable circumstance in which she might be called upon to move.
Definition 4b (formal version 1): A strategy for player i ∈ N in the extensive-form game GE = [N, H, P, f, {Ii}i∈ N, {ui}i∈ N] is a function ai(.) that assigns an action ai(h) to each h ∈ H \ T where P(h) = i with ai(h) = ai(h′) whenever h, h′ ∈ Ii, so that (h, ai(h)) ∈ H.
Definition 4c (formal version 2): Consider the extensive-form game GE = [N, H, P, f, {Ii}i∈ N, {ui}i∈ N]. Let A denote the set of possible actions in the game and A(Ii) the set of possible actions at information set Ii ∈ Ii. A strategy for player i in GE is a function si(.) that assigns an action si(Ii) ∈ A(Ii) to each Ii ∈ Ii.
Extensive-Form and Normal-Form Representation
Definition 5: An outcome in an extensive-form game GE = [N, H, P, f, {Ii}i∈ N, {ui}i∈ N] is one of the terminal histories (in T) of the game.
Note: Payoffs are defined over outcomes. In extensive-form games without chance moves a strategy profile s = (s1,…,sn) results in an unique outcome. Thus, in such games there is an unique payoff vector associated with each strategy profile. In extensive-form games with chance moves a strategy profile s = (s1,…,sn) results in a probability distribution over outcomes. To get to the payoff vector associated with s calculate expected payoffs. This yields again an unique payoff vector for each strategy profile.
From previous discussion it is clear that for any extensive-form game representation of a game, there is an unique normal-form representation (more precisely, it is unique up to any renaming or renumbering of strategies). The converse is not true, however. Many different extensive-
forms may be represented by the same normal-form (see below).
Transforming an Extensive-Form Game without Chance Moves into a Normal-Form Game
6, 2 2, 1
6, 3 6, 3
6, 2 2, 1
6, 3 6, 3
r r
6, 2 4, 5
6, 3 .., ..
.., ..
..., ..
6, 3 6, 3
r l
1, 4 1, 4
1, 4 .., ..
5, 0 5, 0
5, 0 5, 0
l r
1, 4 1, 4
1, 4 1, 4
5, 0 5, 0
5, 0 5, 0
l l
R R R R
R L R
L R R
L L L
R R L
R L L
L R L
L
L
Transforming an Extensive-Form Game with Chance Moves into a Normal-Form Game
.., ..
.., ..
.., ..
.., ..
T
.., ..
.., ..
.., ..
2, 2 H
T T T
H H
T H
H
Transforming a Normal-Form Game into an Extensive-Form Game
2 , 0 -1, -1
r
1 , 1 1 , 1
l
R L
player B
player A
Mixed Strategies
Up to now, we have assumed that players make their choices with certainty. There is no a priori reason to exclude the possibility that a player could randomize when faced with a choice.
Definition 6. Given player i’s (finite) set of pure strategies Si, a mixed strategy for player i, σi : Si→ [0, 1] assigns to each pure strategy si∈ Si a probability σi(si) ≧ 0 that it will be played, where
Notation. ∆(Si), the set of probability distributions over Si with typical element σi, is called the mixed extensions of Si.
Note: If player i has M pure strategies in the set Si = {si1,…,siM} and σim = σi(sim) then ∆(Si) = {(σi1,…, σiM)∈ ℝM: σim ≧ 0 for all m = 1, …,M and ∑Mm=1 σim = 1}
Note: A pure strategy can be viewed as a special case of a mixed strategy in which the probability distribution over Si is degenerate.
Notation: Define σ = (σ1,…, σN ) and σ-i ∈ ∆(S-i) = xj ≠ i ∆(Sj) analogously to the pure strategy case.
Apply definitions to Examples 1 and 2.
( )
=1∑
s∈S i sii
i σ
Mixed Strategies and Expected Utilities
When players randomize over their pure strategies, the induced outcome is itself random. Since each player’s payoff function ui(s) is of the von Neumann – Morgenstern type, player i’s payoff given a profile of mixed strategies σ = (σ1,…,σn) for the n player is her expected utility, the
expectation being taken with respect to the probabilities induced by σ on pure strategy profiles s = (s1,…,sn).
Notation: Ui: ∆(S) ↦ℝis a vNM expected utility function that assigns to each σ ∈ ∆(S) the expected utility (using ui) of the lottery over S induced by σ.
Note: If players mix according to σ then
Notation: The game ΓN =[N, {∆(Si)}i∈N, {Ui}i∈N], where ∆(Si) is the set of probability distributions over Si and where Ui is derived from σ and uias stated above, is called the mixed extension of GN = [N, {Si}i∈N, {ui}i∈N]
∑ ∏
∈ ∈
=
S
s j N
j j i
i u s s
U (σ) ( ) σ ( ).
