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Inductive Game Theory: Notes
@valeria aikat
August 9, 2022
1 Festival Game
A festival game is a motivating example of inductive game theory, which was introduced by
Kaneko and Kimura (1992) and farther analyzed by Kaneko and Matsui (1999). It is intended
to give a theoretical explanation on how phenomena of discrimination and prejudices endoge- nously arise, without assuming individuals’ exogenous incentive scheme for such phenomena.
The key is that individuals do not have prior knowledge of the structure of the society. In- stead, they form their own view of the society depending on experiences. The implication
is that inductively derived views based on individual experiences can lead to discriminatory
behavior and prejudices.1
A festival game itself is a two-stage game. Players are supposed to play the two-stage
festival games repeatedly. The distinct feature is that players’ knowledge on the structure of
the game is not assumed. Instead, after each play, players get information and inductively form
their image of the structure. The festival game is suitable to analyze how players form their
image of the society from their past experiences, when they do not have complete information
on the society.
1.1 Formulation
We introduce the festival game by following Kaneko and Matsui (1999). The construction is
done step by step. First, players of the festival game Γ are introduced as follows:
• The set of players N = {1, ...n};
• The set of ethnicities E = {e1, ..., ey}. The ethnicity function e : N → E assigns the
ethnicity e(i) of each player i;
• The partition of the set of players Ne1
, ..., Ney with #Ne ≥ 2 for e = e1, ..., ey. Note
that i ∈ Ne(i)
;
1The critical difference of prejudices and discrimination is that the former is constructed in individuals’
internal mind, while the latter can be externally observed as behavior. Though discrimination can be expressed
by strategies, prejudices cannot be fully explained without formulation of players’ subjective view on the society.
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The players are supposed to play a certain festival game Γ repeatedly. In each play, they
have opportunity to learn the structure of the game.
A festival game is a two-stage game. In each stage, the set of alternatives for the players
are given as follows:
• The set of festival locations L = {l1, ..., lz}, from which each player chooses where to
hold a festival in the first stage. Player i’s choice in this stage is denoted by fi ∈ L. Let
f = (f1, ..., fn);
• Given f = (f1, ..., fn), player i observes the ethnicity configuration of fi
, which is defined
as the set:
Ei(f) = {e(j) : fj = fi and j 6= i};
• The set of attitude A = {0, 1}, from which each player chooses how to behave in the
festival in the second stage. The friendly and unfriendly attitude is denoted by 1 and 0
respectively. Player i’s choice in this stage depends on the choice of the location in the
first stage and the ethnicity configuration he or she observed. It is formally expressed
by a function ri
: L × 2
E → A.
Following the set of alternatives given as above, a strategy of the players in this game is
constructed as follows:
• A strategy of player i is a pair (fi
, ri). Let ri(f) = ri(fi
, Ei(f)) and r(f) = (r1(f), ..., rn(f));
• Let Σi be the set of all strategies of player i. For a strategy profile σ = (f, r) ∈ Σ ≡
×n
i=1Σi
, the realization path is given by a pair (f, r(f)).
Finally, depending on a strategy profile, the payoff function is given as follows:
• Given each strategy profile σ = (f, r), the payoff function of each player i is given as
follows:
Hi(f, r) = ri(f) · (μi(f, r) − m0),
where the function μi(f, r) is called a mood and given as
μi(f.r) = X
fj=fi,j6=i
rj (f),
and m0 is a noninteger number greater than 2.
Be aware that the payoff function does not include any incentive scheme for discriminatory
behavior of individuals.
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1.2 Knowledge, Experimentations, and Inductive Stability
The important characteristic of the festival game is that, players do not have complete infor- mation about the structure of the game beforehand. To make it clear, the following postulate
is assumed.
Postulate 1. (Knowledge structure)
(1) After each play of game Γ, player i observes only his utility value Hi(σ), if the game is
played according to σ, as well as the information he or she obtained in the play.
(2) Player i knows the set of festival locations L for his or her first choice, and that he or she
has two options, friendly and unfriendly actions (1 and 0) in the festival location he or she
chose.
Players are ignorant of the other structures of Γ, even the set of players N. They get
information from experiences in each play. In general, an experience is a tuple
[fi
, δi
, E; hi
],
where fi and δi are player i’s choice in Γ, E is the ethnicity configuration he or she observed,
and hi
is the utility value he or she obtained.
Next, behavioral patterns of players are explained. In a stationary state, players follow
a fixed strategy profile. However, with a small probability, one of players possibly deviate
from the stationary state for an experimentation. To make it clear, the following postulate is
assumed.
Postulate 2. (Behavior patterns and experimentations)
(a) Given a stationary state σ
?
, each player i behaves according to his behavior pattern σ
?
i
,
subject to stochastic trial deviations with small probabilities once in a while, but after each
trial, he returns to his own behavior pattern σ
?
i
.
(b) Events of trials simultaneously made by two or more players have negligible frequencies,
and they are ignored by the players.
According to its characteristics, individual experiences are divided into three groups: sta- tionary experiences, active experiences, and passive experiences. Stationary experiences are
obtained from the behavior in the stationary state. Active experiences are obtained from
own experimentations. Passive experiences are obtained from other players’ experimenta- tions. Additionally, passive experiences are divided into two groups according to who does
experimentations: an outsider (f
?
j
6= f
?
i = fj ) or an insider (f
?
j = f
?
i
). Formally, each type of
experience is given as follows:
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• The stationary experience for player i under σ
? = (f
?
, r?
) is given as
(S) : s(i|σ
?
) = [f
?
i
, r?
i
(f
?
), Ei(f
?
); Hi(σ
?
)];
• An active experience under σ
?
induced by a trial (fi
, δi) 6= (f
?
i
, r?
i
(f
?
)) of player i is
given as
(A) : [fi
, δi
, Ei(f
?
−i
, fi); Hi(σ
?
−i
,(fi
, δi))];
• A passive experience induced by a trial (fj , δj ) of some outsider j with f
?
j
6= f
?
i = fj is
given as
(PO) : [f
?
i
, r?
i
(f
?
−j
, fj ), Ei(f
?
−j
, fj ); Hi(σ
?
−j
,(fj , δj ))];
• A passive experience induced by a trial (fi
, δi) 6= (f
?
i
, r?
i
(f
?
)) of some insider j with
f
?
j = f
?
i
is given as
(PI) : [f
?
i
, r?
i
(f
?
−j
, fj ), Ei(f
?
−j
, fj )); Hi(σ
?
−j
,(fj , δj ))];
Let the set of all active experiences be A(i|σ
?
). Note that the stationary experience
is not in A(i|σ
?
). Also, let the set of all passive experiences be P(i|σ
?
). Let E(i|σ
?
) =
A(i|σ
?
) ∪ P(i|σ
?
). For convenience, let [φi
; hi
] be a generic element of E(i|σ
?
).
Finally, players’ decision making process is discussed. Depending on new information from
experiences, they may have an incentive to change the frequency of deviation. To make it
clear, the following postulate is assumed.
Postulate 3. (Inductive decision making)
(a) If no active experience in A(i|σ
?
) gives a higher payoff to player i than his stationary payoff
Hi(σ
?
), then he continues behaving according to σ
?
i
(still subject to his occasional trials).
(b) If some active experience [φi
; hi
] in A(i|σ
?
) gives a higher payoff to player i than his
stationary payoff Hi(σ
?
), then he would increase intentionally (maybe slightly, or maybe
drastically) the frequency of the deviation inducing [φi
; hi
].
Following the postulate, the notion of inductive stability is introduced. Inductive stability
is considered as another representation of a Nash equilibrium, following the steady-state
interpretation of a Nash equilibrium.
Definition 1. (Intentional deviation)
Player i has an incentive for an intentional deviation from σ
?
iff there is an active experience
[φi
; hi
] ∈ A(i|σ
?
) with hi > Hi(σ
?
).
Definition 2. (Inductive stability)
A strategy profile σ
?
is inductively stable iff no player has an incentive for an intentional
deviation from σ
?
.
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Proposition 1. A strategy profile σ
?
is inductively stable if and only if it is a Nash equilibrium
in Γ.
Proof. Inductive stability of σ
?
is equivalent to that for any i ∈ N and any [φi
; hi
] ∈ A(i|σ
?
),
Hi(σ
?
) ≥ hi
. This is equivalent to that for any σi ∈ Σi
, Hi(σ
?
) ≥ Hi(σ
?
−i
, σi).
The proposition confirms that the notion of inductive stability is consistent with the
interpretation of a Nash equilibrium as a stationary state.
1.3 Nash Equilibria
In this section, we discuss individuals’ “rational” discriminatory behavior. For the subsequent
analysis, assume the following: for any e ∈ E and any l ∈ L, #{i ∈ Ne : f
?
i = l} 6= 1. The
assumption simplifies the analysis.
Theorem 1. A strategy profile σ
? = (σ
?
1
, ..., σ?
n
) = ((f
?
1
, r?
1
), ...,(f
?
n
, r?
n
)) is a Nash equilibrium
if and only if for any i ∈ N and any e ∈ E,
(a) if μi(σ
?
) ≥ m0, then f
?
j = f
?
i
for any j with e(j) = e and r
?
j
(f
?
) = 1 for any j with
f
?
j = f
?
i
;
(b) if μi(σ
?
) ≥ m0, then μi(σ
?
) ≥ μi(σ
?
−i
(fi
, 1)) for any fi ∈ L;
(c) if μi(σ
?
) < m0, then μi(σ
?
) = 0, i.e., r
?
j
(f
?
) = 0 for any j with f
?
j = f
?
i
;
(d) if μi(σ
?
) < m0, then m0 > μi(σ
?
−i
,(fi
, 1)) for any fi ∈ L.
The proof is given by Kaneko and Matsui (1999) in a straightforward manner. Following
the theorem, there are three types of equilibria:
• Amalgamation equilibria: all players choose the same festival location and act
friendly. These equilibria correspond to special cases of (a) and (b).
• Segregation equilibria: some players of different ethnicities go to different festivals
and at least one festival is active. These equilibria correspond to special cases of (a)
and (b).
• No-festival equilibria: all players choose a festival location arbitrary and act un- friendly. These equilibria correspond to (c) and (d).
The focus is on segregation equilibria. The following corollary characterizes discrimination
behavior in a segregation equilibrium.
Corollary 1. Let σ
? = (f
?
, r?
) be a segregation equilibrium. Suppose f
?
i
6= f
?
j with μi(σ
?
) >
μj (σ
?
), and let fj = f
?
i
.
(a) If μj (σ
?
) ≥ m0, then μj (σ
?
−j
,(fj , 1)) ≤ μj (σ
?
).
(b) If μj (σ
?
) = 0, then μj (σ
?
−j
,(fj , 1)) < m0.
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The intuition of (a) is that, the presence of individual j in f
?
i undermines the mood of
in f
?
i
. Here, discrimination is incurred as a player of a smaller festival f
?
j
comes to a larger,
more active festival f
?
i
. The intuition of (b) is similar: the presence of individual j in f
?
i
,
who came from a completely inactive festival f
?
j
, undermines the mood of in f
?
i drastically.
In both cases, we observe discriminatory behavior incurred by players who regularly go to a
larger, more active festival.
Be noted that, so far, we have not discussed anything on how individuals construct their
own view of the game from their experiences. In the next section, we formalize this part. To
simplify the subsequent analysis, we focus on the following equilibria.
Definition 3. (Fully active equilibria)
a strategy profile is a fully active equilibrium iff it satisfies
FA : r
?
i
(f
?
) = 1, for any i ∈ N.
Everyone acts friendly under FA. Note that condition FA allows only some segregation
equilibria.
1.4 Inductive Construction of an Individual Image
1.4.1 Individual Models
Individuals form their image of the society based on their experiences. Individual views are
summarized as individual models. An individual model is a player’s subjective structure of
Γ.
Definition 4. (Individual models)
An individual model MI of player i is a tuple (N, ˆ Z, ˆ oˆi
, uˆi
; x
0
, X), where
• The set of imaginary players Nˆ;
• The set of potential social states Zˆ;
• The observation function ˆoi
, which is a function on Zˆ;
• The utility function ˆui
, which is a real-valued function on Zˆ;
• The stationary social state x
0
, which is an element of Zˆ;
• The set of relevant social states, which is a subset of Zˆ containing x
0
.
Individual models are not given arbitrarily. The following three assumptions are made on
MI = (N, ˆ Z, ˆ oˆi
, uˆi
; x
0
, X):
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