Game Theory: Lecture 1
Economics 351b: Mathematical Economics: Game Theory
Philipp Strack, Summer 2021
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Introduction
Game theory is a theory of rational behavior of people whose choices, interests are
independent but whose preferences are interdependent.
Its area of application goes considerably beyond games in the usual sense. It applies to
biology, political science, economics, management sciences, operations research, and yes,
actual games, occasionally. Most of the applications in this course will be drawn from
economics.
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How to represent a game
2.1
Motivation
Here is the verbal description of three games:
• The instructor hides a dice in one of two glasses, without the student observing his
choice. The two glasses are presented, upside down, on the table. The student wins
$1 (from the instructor) if he points to the glass containing the dice. Otherwise, the
instructors wins $1 (from the student). The two glasses are opaque.
• The instructor hides a dice in one of two glasses, without the student observing his
choice. The two glasses are presented, upside down, on the table. The student wins
$1 (from the instructor) if he points to the glass containing the dice. Otherwise, the
instructors wins $1 (from the student). The two glasses are transparent.
• The instructor and the student simultaneously announce “Left” or “Right.” The
student wins $1 (from the instructor) if the two announcements coincide. Otherwise,
the instructors wins $1 (from the student).
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I suspect that most of you realize that the first two games are not equivalent, although
they look a lot alike. Most of you, also, probably feel that the first and third game (commonly known as matching pennies) have a lot in common, although the descriptions
differ quite a bit (in particular, in the first one, choices are made sequentially; in the last,
they are made simultaneously).
We need to develop a formal definition and representation of games that allows
us to distinguish those two situations. This definition should be such that games that are
strategically equivalent, as the first and last game appear to be, are treated identically,
while games like the first and the second one should be distinguished. This is the topic
of this first set of lecture notes.
As we shall see, there are several ways of representing a game. Depending on the
game, one or the other might turn out to be more convenient. Nevertheless, it is always
possible to use the following, graph-theoretic, representation. First, we need a way to
represent the sequence of moves. This is done using game trees.
2.2
Game trees
We start with the notion of a directed graph.
Definition 2.1 A directed graph is a finite set of points (called vertices, or nodes)
together with a set of directed line segments (called edges, or branches) between some
pairs of distinct vertices
Given some directed graph G, let X(G) denote the set of its vertices, or simply X if there
is no risk of confusion, and let E(G), or E, denote its edges.
Definition 2.2 A path from a vertex x to a vertex x′ in a directed graph G is a finite
sequence (x0 , . . . , xn ) n ≥ 1, with x0 = x and xn = x′ and for all i ≥ 1, (xi−1 , xi ) an edge
in G.
Definition 2.3 A tree is a directed graph T if
1. ∃ a distinguished vertex r (the root) with no edges going into it;
2. for every other vertex x, ∃ a unique path from r to x.
Let T be a tree. A node x′ is a child of x ∈ X(T ) if (x, x′ ) is an edge. The node x is
then the parent of x′ . A terminal node is a node with no children. Let Z denote the
set of terminal nodes. A node that is not a terminal node is a decision node. A node x′
is a descendant of x if there is a path from x to x′ . In this case, x is an ancestor of x′ .
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The length of a path is the number of edges in it. The depth of a tree is the length of
the longest path in it. The subtree determined by x, denoted Tx , is defined as follows:
The nodes of Tx are x itself plus all of the descendants of x. The edges of Tx are all edges
of T which start at a vertex of Tx .
The proof of the following observations are left as easy exercises. Let T be a tree.
1. No node x ∈ X(T ) has more than one parent.
2. If x, x′ ∈ X(T ) and there is a path from x to x′ , then there is no path from x′ to x.
3. Every decision node has at least one terminal descendant.
2.3
The extensive form
Having a tree is not enough. What we further need to know is:
1. The set of players.
2. The order of moves: who moves when.
3. What the players’ choices are when they move.
4. What each player knows when he makes his choices.
5. The probability distribution over exogenous events.
6. The players’ payoffs as a functions of all the choices and events.
The penultimate point needs some additional explanation. Consider, for instance,
playing at a slot-machine. It makes sense to model this as a game with only one player,
the gambler. But his payoff depends also on something else, namely chance, often referred
to as Nature in game theory, or player 0. Nature is a very special, additional, player,
without preferences, and whose way of playing must be specified to begin with. Hence,
we must specify the probability distribution over exogenous events.
All this information is provided by the extensive form. Recall that a partition of a
finite set B is a collection of disjoint subsets of B whose union is B.
Definition 2.4 An n-person game in extensive form is a tuple Γ := (T, P, H, C, p, u)
where:
1. T is a finite tree, with nodes X and edges E.
2. The player partition: P := {P0 , P1 , . . . , Pn } is a partition of X \Z; this indicates,
for each decision node, whose turn it is to play.
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3. The information partition: H = {H1 , . . . , Hn }, where Hi is a partition of Pi ,
for i = 1, . . . , n, such that no path in T intersects some element h ∈ Hi twice, and
if x, x′ ∈ h ∈ Hi , then x and x′ have the same number of children; this indicates
decision nodes that a player cannot distinguish.
4. The choice partition: C is a partition of E such that, ∀i = 1, . . . , n, if (x, x′′ ) ∈
c ∈ C, with x ∈ h ∈ Hi , then ∀x′ ∈ h, there exists a unique x′′′ such that (x′ , x′′′ ) ∈ c;
these are the choices available to the players.
5. The probability assignment: p := {px : x ∈ P0 }, where px is a probability
distribution over the children of x; this indicates how Nature plays at each of its
decision nodes.
6. The utilities: u := {u1 , . . . , un }, where ui : Z → R; this indicates, for each
terminal node, the payoff of each player.
Choices are also called actions, or moves, and often denoted ai (for actions of player i).
We write Ch , or A(h), for the choices available at h ∈ Hi .
Remark 2.5 The condition which we imposed on the choice partition ensures that for
every point x in an information set there is a unique outgoing edge associated with a
certain action. If some action is associated with the edge (x, x′′ ) then the condition implies
that
1. that no other edge (x, x′′′ ) starting in the same node is associated with the same
action as we required uniqueness; and
2. that for every other x′ 6= x in the same information set h there exist an edge associated with the same action to make sure that the game tree describes at every node
in that information set how play continues when that action is take.
The last item is often replaced by a rational preference relation %i for each player
i (a complete, transitive binary relation) over terminal nodes: if z, z ′ ∈ Z, z %i z ′ means
that i prefers z to z ′ (weakly). We write ≻i for the strict version, i.e. z ≻i z ′ if z %i z ′ ,
but not z ′ %i z. We write z ∼i z ′ if both z %i z ′ and z ′ %i z, so that player i is indifferent
over z and z ′ . Rational preference relations are in a sense more general than utilities
(every utility vector defines a preference relation, but not conversely), and we shall work
with those as long as possible.
Almost all games in the literature (and all of those we will talk about) are games with
perfect recall. No player ever forgets what he once knew, and what he once did. We have
already required that no path intersects the same information set twice, but this is not
enough. Formally:
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Definition 2.6 A game has perfect recall if whenever x′ , x′′ ∈ h ∈ Hi , and x ∈ Pi is
an ancestor of x′ , then ∃x̂ ∈ h′ ∈ Hi , such that (i) x ∈ h′ , (ii) x̂ is an ancestor of x′′ ,
(iii) the choice at x̂ on the path (x̂, x′′ ) is the same as the choice at x on the path (x, x′ ).
Intuitively, if player i cannot distinguish two nodes, he must have played the same
way to reach each of them. Henceforth, we also implicitly assume that the games we are
dealing with are games with perfect recall.
An important class of games are games of perfect information.
Definition 2.7 An extensive game Γ is a game with perfect information if, ∀i =
1, . . . , n, ∀h ∈ Hi , h is a singleton.
An extensive form game without perfect information is said to have imperfect information.
Checkers and chess have perfect information. Bridge and poker do not (and they also
have chance). Matching pennies has no chance, but no perfect information either.
Here are the extensive forms representing the first three games that we have discussed.
The first and third game are represented on the left, the second game is represented on
the right. By convention, information sets are represented by dotted lines. Also, payoffs
are represented by a vector. By convention, the first entry is the payoff of player 1, which
in turn is typically identified with the first player to move. Observe that extensive form
trees cannot capture simultaneous moves. We model them by assuming that they are
taken sequentially, but that the players moving later do not observe the earlier choices.
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L1
1
R1
L1
R1
2
L2
−1, 1
R2
1, −1
L2
1, −1
R2
L2
−1, 1
−1, 1
R2
1, −1
L′2
1, −1
R2′
−1, −1
Figure 1: Game forms for the three examples.
Here is another example, Kayles: There are four skittles, and players remove 1 or 2
adjacent skittles in turn. The loser is the player who picks the last skittle. There are two
outcomes: W (Win) for player 1, or L (Loss) for player 1. We first represent the game as
it unfolds.
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✲
L
1
✻
W ✛
L ✛
✛
✻
✛
2
✲
1
1
✻
2
✛
✲
r
1
r r
2
r
1
✲
2
W
L
1
✲
W
✲
L
✲
W
2
✲
1
❄
W
✛
✛
✛
2
1
❄
✲
2
r
✲
❄
L
1
❄
L
Figure 2: Description of Kayles.
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2
1
a′1
a1
a′′1
2
a2
L
a′2
a′′′
1
2
a′′2
b2
L
1
W b1
L
2
b′2
c2
1
b′1
L
W
Wc1
L
c′2
c′1 W
c′′2
L
W
Figure 3: Game form of Kayles.
However, by convention, we do not represent forced moves (i.e., we skip steps in which
a player has only one choice). This gives the following extensive-form game.
2.4
What’s a strategy?
We have now described what players can do. Let us talk about what they plan to
do. We must introduce here the notion of a strategy. A strategy for a player is a plan of
action that is complete and respects information sets. Let Ai := ∪h∈Hi A(h).
Definition 2.8 A pure strategy si for player i is a map si : Hi → Ai such that ∀h ∈ Hi ,
si (h) ∈ A(h).
The set of all strategies for player i in a given game Γ is denoted Si := ×h∈Hi A(h). A
pure strategy profile is a vector of strategies s := (s1 , . . . , sn ). Let S := ×i Si .
In the game of Kayles, for instance, a pure strategy for player 1 is a vector in
′
′
′
′
{a1 , a′1 , a′′1 , a′′′
1 } × {b1 , b1 } × {c1 , c1 }; for instance, (a1 , b1 , c1 ) is a pure strategy. (Note
that a strategy defines what a player does even at information sets that are precluded by
what his strategy specifies at earlier information sets.)
For reasons that will become clear later on, it is also important to let players make
unpredictable, i.e. random choices (think about poker). There are two ways to define
such random choices, called mixed and behavior strategies. We will come back to the
first one later. The second one is defined as follows. Given any finite set B, let △B define
the probability distributions (also called lotteries) over
P B. More formally, if |B| denotes
|B|
the number of elements in B, then △B := {p ∈ R | j∈B pj = 1, and pj ≥ 0, all j}.
Definition 2.9 A behavior strategy for player i is a map σi : Hi → △Ai such that
∀h ∈ Hi , σi (h) ∈ △A(h).
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So it is simply an element of Σi := ×h∈Hi △A(h). A behavior strategy specifies a probability distribution at each information set, and the distributions at different information sets
are independent. Observe that a pure strategy is just a special case of a behavior strategy.
A behavior strategy profile is a collection σ := (σ1 , . . . , σn ), and let Σ := ×i Σi . Any behavior strategy profile generates recursively a probability distribution over terminal nodes
Z.
In the case of Kayles, for instance, a behavior strategy for player 1 is then simply a
′
triple of probability distributions, that is an element of △{a1 , a′1 , a′′1 , a′′′
1 } × △{b1 , b1 } ×
△{c1 , c′1 }.
Observe that, if Nature is not a player, we could immediately have defined the game by
specifying simply the sets Si , and functions ui : S → R, i = 1, . . . , n, by defining ui (s) as
the payoff obtained from the terminal node that is reached under the pure strategy profile
s. (Even if Nature was a player, we could do so, by computing the relevant expectations.)
This leads to the following definition.
Definition 2.10 A normal-form game is a collection (Si , ui )i=1,...,n , where Si is a
finite set and ui : ×Si → R.
We write henceforth for short (S, u) for the normal form, where S = ×i Si , u = (u1, . . . , un ).
Again, a more general version of the definition uses preference relations %i , all i, instead
of utility functions.
In general, in a given extensive form, there are many strategies that are equivalent, in
the sense that, no matter the strategy profile used by the other players, those strategies
induce the same distribution over terminal nodes z ∈ Z. For instance, this is often the
case if a given player gets to move twice along some path: suppose that player i chooses
first between ai and a′i , and later, if he chooses ai , he must choose between bi and b′i . If
his strategy calls for a′i at the first information set, then what he would do at the second
had he chosen ai instead is irrelevant to the outcome of the game, given the strategies of
the other players. (Nevertheless, it is important when we solve extensive form games to
specify strategies fully, because whether the strategies of the other players are optimal,
in the sense to be defined later, can only be evaluated relative to the entire strategy of
player i.) To simplify the analysis in the normal form, we sometimes eliminate all but
one of these equivalent strategies, which leads to the following concept.
Definition 2.11 The reduced normal form of an extensive-form game is obtained
by identifying equivalent pure strategies (i.e., by eliminating all but one member of each
equivalence class).
Observe that there is another way of defining random choices: a player might choose a
probability distribution over Si , that is an element li ∈ △Si . This is a more complicated
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object. In the example of Kayles, the set of such strategies for player 1 is an element
′
′
in △ ({a1 , a′1 , a′′1 , a′′′
1 } × {b1 , b1 } × {c1 , c1 }), a much larger set than the set of behavior
strategies. Formally:
Definition 2.12 A mixed strategy for player i is a probability distribution over Si , that
is, an element li ∈ △Si .
Let Li denote the set of all mixed strategies in a given game Γ in extensive form. Clearly,
Σi ⊂ Li , so that the set of mixed strategies is larger than the set of behavior strategies. It
is then natural to ask whether these two ways of defining random choices are equivalent.
We first need to be precise about what equivalent means.
Definition 2.13 Let Γ be an extensive form game. The strategies σi ∈ Σi and li ∈ Li
are realization-equivalent if, for every profile l−i ∈ L−i , (σi , l−i ) and (li , l−i ) induce
the same distribution over terminal nodes Z.1
Since Σi ⊂ Li , for every σi ∈ Σi , we can find a mixed strategy li such that li are σi are
realization-equivalent. The converse is not obvious, and only holds if the game has perfect
recall. We omit the proof.
Theorem 2.14 (Kuhn theorem) Let Γ be an extensive form game of perfect recall.
Let li ∈ Li be a mixed strategy. Then there is σi ∈ Σi such that li and σi are realizationequivalent.
Because of this theorem, and our focus on games with perfect recall, we need not worry
about the distinction. We shall typically use behavior strategies, σi ∈ Σi , since they are
simpler, but will typically refer to them as mixed strategies.
The simplest class of games are the strictly competitive games, and these are the
games with which we will start our analysis.
Definition 2.15 A strictly competitive game is a two-player extensive-form game
such that
∀z, z ′ ∈ Z, z ≻1 z ′ ⇔ z ′ ≻2 z.
In essence, this makes the preferences “one-dimensional.” Kayles is an example of strictly
competitive game. Chess is another.
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The notation −i refers to all players but i; e.g., if n = 4, −2 = {1, 3, 4}.
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