Coordination Games, Multiple Equilibria and
the Timing of Radio Commercials
Andrew Sweeting
Department of Economics
Northwestern University
27 September 2004
Motivation
• common applied micro problem: observe agents choosing the same action and
want to understand how much of this is explained by interactions between the
agents
• examples include peer effects, social learning, demand/network externalities,
industrial location/agglomeration, social conventions
• models where interactions are important often have multiple equilibria, typically
viewed as creating an estimation problem
• I will show how to use multiple equilibria, both in a simple model and in the
data, to identify the parameters of interest
The Timing of Radio Commercials and Why We Care
• commercial stations sell audiences of commercials to advertisers who only value
people listening to the commercials (2000 revenues c.$20 billion)
• but, many listeners avoid commercials by switching between stations e.g., 70%
in-car, 41% at-home and 29% at-work switch stations at least once during
commercial break
• if stations played commercials at the same time avoidance might be reduced
“What can be done to enhance the probability that listeners really listen? Shorter
breaks would minimize the probability of people leaving the station...The other
option would be transmit breaks at universally agreed, uniform times. Why tune
to other stations if it’s certain that they will be broadcasting commercials as
well?” (Alan Brydon, Campaign, 1994)
Simple Model where Stations Coordinate on Timing
• N stations, N units of listeners: any station is “first choice” of 1 unit of listeners and “second choice” of a (different) 1 unit of listeners; station preferences
independent
• behavior: proportion 1−s of listeners (“non-switchers”) listen to the first choice
independent of what it plays, proportion s (“switchers”) listen to first choice
if it has no commercials or if both first and second choice have commercials,
otherwise they listen to second choice
• audience of a commercial break on i when N−i other stations play commercials
1−s+s
N−i
N −1
• stations maximize audience of commercials by playing them at same time as a
higher proportion of other stations
Simple Model where Stations Play Commercials at Different Times
• 2 stations and a non-commercial outside option, 2 units of listeners, sequence of
discrete time intervals, each station plays music-commercial-music-commercial
• behavior: when commercials play on a station, 12 previous listeners switch to
other station iff it is playing music and otherwise switch to outside option
[returning when commercials are over]; when both stations play music listeners
split equally between them
• station maximizes audience of its commercials
• outcomes: play commercials at same time: audience of each commercial break
1 , play commercials at different times: audience of each commercial break 2
2
3
Empirical Questions
1. do radio stations tend to play commercials at the same time?
2. how much of this pattern is due to “coordination” on choosing same times
rather than “common factors” which make certain times more attractive for
commercials independent of what other stations choose?
3. given widespread avoidance of commercials, why is coordination on the timing
of commercials imperfect?
Histograms of the Number of Stations Playing Commercials Each Minute 12-1pm and 5-6pm
(a) 12-1pm
Nu m b er o f statio n -h o u rs p layin g
co m m ercials in m in u te
20000
15000
10000
5000
0
:00
:05
:10
:15
:20
:25
:30
:35
:40
:45
:50
:55
M inute
Num ber o f station-hours playin g
com m ercials in m inu te
20000
(b) 5-6pm
15000
10000
5000
0
:00
:05
:10
:15
:20
:25
:30
:35
:40
:45
:50
:55
M inute
Note: based on airplay data (described in Section 6.1) from 1,063 contemporary music stations in 146 metro-markets. 12-1pm histogram
based on 48,889 station-hours with commercials at some point in hour, 5-6pm histogram based on 48,567 station-hours with commercials at
some point in hour.
Empirical Strategy
• data on timing of commercials by stations in 146 local radio markets
• estimate a simple coordination game: allow each station to want to coordinate with other stations in its market, assume “common factors” same across
markets and markets independent repetitions of game
• multiple Nash equilibria if and only if incentive to coordinate strong enough
• use multiple equilibria in data (e.g., Boston 5:50pm, SF 5:55pm) and degree
of coordination in each market to identify incentive to coordinate
(a) payoffs
independent
A
B
A
2,2
2,1
B
1,2
1,1
(b) payoffs +0.5
if same action
A
B
A 2.5,2.5
2,1
B
1,2
1.5,1.5
(c) payoffs +2
if same action
A
B
A
4,4
2,1
B
1,2
3,3
Distribution of Commercials, 5-6pm, 29 October 2001
(a) Orlando, FL
5
4
3
2
1
0
0
5
10
15
20
25
30
35
40
45
50
55
35
40
45
50
55
(b) Rochester, NY
5
4
3
2
1
0
0
5
10
15
20
25
30
Existing Literature
1. TV timing: e.g., Epstein (1998), Zhou (2004, forthcoming)
2. empirical “interactions” literature: no direct use of multiple equilibria in model
but use intuition of excess clustering of same choices within groups
3. IO literature on structural models of e.g., firm entry
• recognize multiple equilibria as feature of model but view as econometric problem because gives inequalities for likelihood of some outcomes
• solved by: (a) eliminate multiplicity e.g., sequential structure, (b) use features
common across equilibria, (c) Tamer (2002, 2003): use inequalities
4. large experimental literature on coordination games and equilibrium selection
Summary of Empirical Results
1. evidence of multiple equilibria, allowing incentive to coordinate to be identified,
during drivetime hours; no evidence outside drivetime. Consistent with more
incentive to coordinate during drivetime when there are more in-car listeners
who are more likely to switch.
2. only modest coordination in Nash equilibrium but estimates imply almost perfect coordination with joint payoff maximization
• explained by interaction of (i) difficult for a station to time its commercials at
a precise scheduled time, (ii) externalities in coordination
3. more coordination in medium-sized and smaller markets than largest markets;
and some evidence that stations in geographically close markets tend to choose
same times for commercials
Overview of the Rest of the Talk
2. imperfect information coordination game with multiple equilibria
3. identification
4. estimation
5. testing for multiple equilibria
6. application: the timing of commercials
(a) data
(b) relation of the timing problem to game
(c) results and extensions
7. conclusions
2
Model of a Symmetric, Discrete Choice, Incomplete Information Coordination Game
• players indexed by i = 1, .., N , simultaneously choose one of two actions,
t = 1, 2; in the application, players are stations and t are different times for
commercials
• i’s payoff from choosing action t is:
π it = β t + αP−it + εit
(1)
• i’s strategy (Si) depends on εi1, εi2 and, if α 6= 0, the strategies of other
players (S−i)
• optimal strategy: i will choose action 1 if and only if
β 1 + αE(P−i1|S−i) + εi1 ≥ β 2 + αE(P−i2|S−i) + εi2
(2)
Player i’s payoff from choosing action t
π
it
= β
t
i’s payoff from
choosing t
determines average
payoff of t
independent of other
players’ choices
+ α P − it + ε
“incentive to
coordinate”
α≥0
proportion of
other players
choosing t
it
idiosyncratic
error, IID
across players
and actions,
private
information
assume Type I
extreme value,
(“logit”) mean
zero, scale
parameter 1
• threshold crossing model so normalize β 2 = 0, label β 1 as β; given extreme
value form of ε, the probability i chooses action 1, p∗i , is
eβ+αE(P−i1|S−i)
∗
pi = β+αE(P |S )
−i1 −i + eα(1−E(P−i1 |S−i ))
e
(3)
• if, α ≥ 0, all Bayesian Nash equilibria are symmetric and satisfy
∗
eβ+αp
∗
p =
∗
∗
eβ+αp + eα(1−p )
(4)
• multiple equilibria (up to 3 p∗ satisfy (4))
• proportional formulation (P−it) makes all equilibria independent of number of
players (N )
• analysis of how strategies and multiple equilibria vary with β and α
Comparison of Nash Equilibrium and Expected Joint Payoff
Maximizing Strategies
• in Bayesian Nash equilibrium, imperfect coordination because of the εs and
ignore positive externality on other players choosing same action
• straightforward to show that with expected joint-payoff maximization
JP
eβ+2αp
∗
eβ+αp
JP
∗
p
=
cf. p =
∗
∗
JP )
JP
2α(1−p
β+2αp
eβ+αp + eα(1−p )
e
+e
i.e., coordination “twice” as important as in Nash equilibrium
• can imply large differences in strategies
(5)
Empirical Model & Equilibrium Selection
• data from independent repetitions of game (m = 1, .., M ) giving number of
players choosing actions 1 and 2 (nm1, nm2)
• assume (β, α) constant across repetitions
• if (β, α) support one equilibrium p∗(β, α) then probability of (nm1, nm2) observation
Pr(nm1, nm2|β, α) = Cp∗(β, α)nm1 (1 − p∗(β, α))nm2
(6)
A indicator for repe• if (β, α) support two stable equilibria, then if observed Zm
tition m in equilibrium A
A) = C
Pr(nm1, nm2|β, α, Zm
Ã
A p∗ (β, α)nm1 (1 − p∗ (β, α))nm2 +
Zm
A
A
A
∗
n
(1 − Zm)pB (β, α) m1 (1 − p∗B (β, α))nm2
!
A : treat as “mixture model”
• do not observe Zm
A is inde• suppose reduced form “equilibrium selection mechanism” where Zm
pendent of Nm and εs
A ∼ Bernoulli(λ)
Zm
(7)
• λ to be estimated; it can also depend on observables. Incomplete data probability
Pr(nm1, nm2|β, α, λ) = C
Ã
λp∗A(β, α)nm1 (1 − p∗A(β, α))nm2 +
(1 − λ)p∗B (β, α)nm1 (1 − p∗B (β, α))nm2
• this is pmf of a binomial mixture model with two components
!
(8)
3
Identification
• parameter space: (β, α, λ) with −∞ ≤ β ≤ ∞, α ≥ 0, 0 ≤ λ ≤ 1
• sample space: set of possible outcomes (nm1, nm2), nm1 ≥ 0, nm2 ≥ 0
• define µN as the proportion of observations where nm1 + nm2 = N
• p∗A, p∗B with p∗A ≥ p∗B defined as solutions to:
∂
β+αp∗
e
p∗ =
∗ ) where
∗
α(1−p
β+αp
e
+e
µ
¶
∗
β+αp
e
∗
∗
eβ+αp +eα(1−p )
∂p∗
A ∼ Bernoulli(λ)
Zm
Pr(nm1, nm2|β, α) = C
Ã
< 1 (stable equilibria)
A p∗ (β, α)nm1 (1 − p∗ (β, α))nm2 +
Zm
A
A
A
∗
n
(1 − Zm)pB (β, α) m1 (1 − p∗B (β, α))nm2
(9)
(10)
!
(11)
Definition (Parameter Vector Identification). A parameter vector (β, α, λ) is
identified if and only if for any vector (β 0, α0, λ0),
Pr(nm1, nm2|β, α, λ) = Pr(nm1, nm2|β 0, α0, λ0) ∀nm1, nm2
implies that (β, α, λ) = (β 0, α0, λ0)
(12)
Proposition 1. Parameter vectors (β, α) where (β, α) support only one equilibrium are not separately identified
Intuition: see diagram
Proposition 2. Parameter vectors (β, α, λ) where (β, α) support two distinct
equilibria are identified under two additional conditions:
Condition 1.
0<λ<1
0
Some observations are generated from each equilibrium, i.e.,
P∞
Condition 2. Some repetitions have nm1 +nm2 ≥ 3 players, i.e., j=3 µj >
Intuition: see diagrams
Identification of p*A,p*B from the data
0.25
white columns
pmf for p*=0.5
0.2
0.15
black columns
pmf for p*A =0.7, p*B =0.3,
λ=0.5
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
Comments on Identification Results
• any equilibrium A is supported by many different sets of parameters, but each
of these gives a different equilibrium B
• identification of mixture of different equilibria in the data treated as a standard
mixture problem
• extension to more than two choices e.g., T actions, T − 1 β ts, 1 α:
— if ε Type I extreme value, only need to identify two equilibria to identify all
of the βs and α
— if there are ET possible equilibria need repetitions with 2ET − 1 players to
identify strategies
• identification results extend to cases where parameterize (β, α, λ): see application
4
Estimation (Outline)
• EM Algorithm used to find Maximum Likelihood estimates. The most computationally efficient approach to estimation when allow for two equilibria follows
the identification proof:
1. use EM to estimate
mixing parameter λ
³
´
c
d
∗
∗
b
pA, pB , λ i.e., the two equilibrium strategies and the
b and α
b by solving
2. find β
⎛
∗
pc
A
⎞
³
´
c
∗
b +α
⎠ =β
b 2pA − 1
ln ⎝
c
∗
1 − pA
⎛
d
∗
p
B
⎞
³
´
d
∗
b +α
⎠=β
b 2pB − 1 (13)
ln ⎝
d
∗
1 − pB
d
∗ ,p
∗
b α)
b support pc
3. check that (β,
A B as stable equilibria (paper describes what I
do if they do not)
5
Testing for Multiple Equilibria
• need to test for multiple equilibria because identification depends on having
multiple equilibria in the data
• equivalent to testing whether a binomial mixture model has one or two components
• likelihood ratio test statistic (LRTS) has a non-standard distribution under the
null of a single component/equilibrium
• Chen and Chen (2001) show valid to use parametric bootstrap for LRTS to
test for the homogeneity of binomial mixtures
6
The Timing of Radio Commercials: Data
• panel of airplay logs for 1,063 commercial stations which are home to 146 local
Arbitron-defined radio metro-markets (example)
• 7 music categories: Adult Contemporary, Album Oriented Rock/Classic Rock,
Contemporary Hit Radio/Top 40, Country, Oldies, Rock, Urban; the stations
in my sample are well-established with relatively large listenerships; all stations
treated symmetrically in estimation
• panel data: Monday-Friday, first week of each month in 2001, maximum of 60
days per station
• drivetime vs. non-drivetime comparison: 4 afternoon drivetime hours, 4 other
hours (12-1pm, 9-10pm, 10-11pm, 3-4am)
Mediabase 24/7 - WROR-FM - Thursday, December 06, 2001
Page 10 of 14
3:34 PM FREE
All Right Now
1970
G
3:40 PM WRIGHT, GARY
Dream Weaver
1976
G
3:44 PM JOPLIN, JANIS/BIG BR
Piece Of My Heart
1968
G
3:48 PM JOHN, ELTON
Saturday Night's Alright For..
1973
G
3:54 PM EAGLES
Please Come Home For Christmas
1978
G
4 PM
Stop Set BREAK
Commercials And/Or Recorded Promos
4:02 PM WHO
Behind Blue Eyes
1971
G
Have You Ever Seen The Rain
1971
G
4:10 PM JOEL, BILLY
My Life
1978
G
4:12 PM DYLAN, BOB
Like A Rolling Stone
1965
G
4:18 PM MCCARTNEY, PAUL
Maybe I'm Amazed
1970
G
You Ain't Seen Nothing Yet
1974
G
4:26 PM WAR
Low Rider
1975
G
Stop Set BREAK
Commercials And/Or Recorded Promos
4:38 PM SPRINGSTEEN, BRUCE
Born To Run
1975
G
4:42 PM CROSBY, STILLS, & NA
Just A Song Before I Go
1977
G
4:44 PM JEFFERSON AIRPLANE
Somebody To Love
1967
G
4:48 PM LOGGINS & MESSINA
Your Mama Don't Dance
1972
G
Stop Set BREAK
Commercials And/Or Recorded Promos
4:58 PM HOLLIES
Long Cool Woman (In A Black..)
1972
G
5:00 PM CLAPTON, ERIC
Cocaine
1980
G
5:04 PM BEATLES
While My Guitar Gently Weeps
1968
G
5:08 PM GRAND FUNK
Some Kind Of Wonderful
1974
G
5:12 PM TAYLOR, JAMES
Carolina In My Mind
1976
G
4:06 PM
4:22 PM
CREEDENCE
CLEARWATER
BACHMAN-TURNER
OVERD
5 PM
file://C:\Andrew\MIT H Drive\Radio\MultipleEquilibriaPaper\Paper\Slides\WROR-FM6-...
9/22/2004
CUME DUPLICATION (Home to Market Boston Music Stations, Fall 2002)
Number in (row,column) gives the percentage of listeners to row station who also recorded listening to column station
W
Q
S
X
Y
W
M
J
X
Y
W
X
R
V
Y
W
B
O
S
Y
W
B
C
N
Y
W
B
M
X
Y
W
F
E
X
N
W
F
N
X
Y
W
R
O
R
Y
W
Z
L
X
Y
W
A
A
F
Y
W
X
K
S
Y
W
O
D
S
Y
W
K
L
B
Y
Rock
CHR
Olds
Cntry
W
B
O
T
Y
W
I
L
D
N
W
J
M
N
Y
Station
Specific Format
WMJX-FM
AC
-
16
12
5
11
21
0
3
16
13
2
30
15
8
7
2
23
WQSX-FM
AC
27
-
16
3
18
32
1
8
11
17
7
50
14
7
10
1
38
WBOS-FM
AAA
21
17
-
26
27
32
1
9
17
31
5
27
13
4
2
0
10
WXRV-FM
AAA
15
6
48
-
24
18
1
7
11
13
4
13
9
5
48
0
6
WBCN-FM
Alternative
11
11
16
7
-
18
1
16
13
23
18
26
13
4
4
1
19
WBMX-FM
Alternative
24
22
21
6
20
-
1
8
13
19
7
45
13
6
3
0
25
WFEX-FM
Alternative
15
23
20
13
25
37
-
0
30
50
14
45
6
0
0
0
29
WFNX-FM
Alternative
10
15
17
8
51
22
0
-
11
28
27
33
9
3
5
1
27
WROR-FM
Clsc Hits
25
11
15
5
21
18
1
6
-
30
7
20
32
8
15
0
9
WZLX-FM
Clsc Rock
16
12
22
9
28
20
1
10
23
-
9
23
19
7
22
0
16
WAAF-FM
Rock
6
11
8
3
45
15
1
20
11
18
-
29
8
5
3
0
26
WXKS-FM
CHR
22
23
12
3
20
30
1
8
9
14
9
-
10
6
12
1
39
WODS-FM
Oldies
17
10
9
3
15
13
0
3
23
18
4
16
-
11
2
1
12
WKLB-FM
Country
15
9
5
3
8
11
0
2
10
12
4
17
19
-
1
1
10
WBOT-FM
Urban Contemporary
22
19
3
1
11
8
0
5
2
4
4
24
6
2
-
16
79
WILD-AM
Urban AC
17
7
1
0
5
2
0
2
2
1
9
4
3
45
-
33
WJMN-FM
Urban CHR
19
19
5
2
16
18
1
7
5
11
43
8
4
21
3
-
AC
AAA
Alternative
Clsc Rock
9
Urban
Coverage of the Airplay Sample
Metro-markets
Metro-markets
1(New York, NY)- 71(Knoxville, TN)
70(Ft. Myers, FL)
and above
# markets
# rated music
mean
stations (home)
# airplay
mean
stations (home)
Airplay stations’
mean
proportion
of rated listenership
69
77
14.6
9.7
10.1
4.7
0.833
0.671
(b) Intepretation of the Payoff Function
• treat time as discrete intervals, in which stations either have commercials or
play music
• interpretation of the payoff function: station i in market-day m in hour h payoff
from commercial in interval t
π imht = β ht + αhP−imht + εimht
(14)
β ht: some intervals more attractive for commercials, same across stations and
markets in hour h;
αhP−imht: payoff increasing with the proportion of other stations in mh with
commercial in t (symmetry), αh same across stations and markets in h;
εimht: private information IID component of timing preferences - two sources
WROR (Boston)'s Timing of Commercials in the First Week of November 2001, 5-6pm
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
Monday
Tuesday
Wednesday
Thursday
Friday
Warren (2001), p.24: sweeping the quarter-hours with music "can be done some of the time. But it can't
be done consistently by very many stations. Few songs are 2:30 minutes long any more."
Definition of a Time Interval (t), Players
• 5 minute discrete time intervals (:03-:07,:08-:12,..)
• identify median minute of each break from log and impute each break to exactly
one time interval
• here two simplifications: (i) focus on just two time intervals : 48-:52 (“interval
1”) and :53-:57 (“interval 2”); (ii) for basic model: players are stations with
exactly one break in one of these intervals: equivalent to assuming expected
probability of stations having breaks same across markets
• note: also find “excess clustering” of commercials within markets relative to
timing across markets, especially during drivetime, if use minute-by-minute
data across the whole hour and control for the music category x quantity of
commercials during hour x hour of day on each station
Proportion of station-hours with commercial breaks in the 2 time intervals
# station Interval 1 only
-hours
:48-:52
3-4 am
39,317
0.248
12-1 pm
48,889
0.311
3-4 pm
49,716
0.332
4-5 pm
49,331
0.317
5-6 pm
48,567
0.326
6-7 pm
49,015
0.336
9-10 pm
46,329
0.283
10-11 pm
45,074
0.284
Interval 2 only
:53-:57
0.198
0.281
0.267
0.290
0.306
0.258
0.304
0.291
(based on station-hours with at least one commercial during hour)
Both
1&2
0.002
0.002
0.001
0.002
0.001
0.001
0.001
0.001
Empirical Results: Basic Model
• each market-day is a separate and independent observation of the game and
the game is estimated separately for each hour
(i) Evidence of Multiple Equilibria (LRTS) - Table 4 in paper
Afternoon drivetime hours
Non-drivetime hours
3-4pm
4-5pm
5-6pm
6-7pm
3-4am
12-1pm
9-10pm
10-11pm
14.6
18.2
11.5
8.1
0
0
0.8
0
<0.2%
<0.2%
<0.2%
0.6%
-
-
21.2%
-
Number of market-days
7,616
7,672
7,750
7,654
6,551
7,616
7,528
7,405
Number of station-days
29,752
29,968
30,739
29,187
17,551
28,993
27,246
25,957
LRTS
Significance level
based on bootstrap
(ii) Estimates of (β, α, λ) and Equilibrium Strategies for Drivetime Hours
β
α
λ
Nash equilibrium
* *
strategies (pA,pB )
Joint Expected Payoff
3-4pm
4-5pm
5-6pm
6-7pm
0.001 (0.002)
0.001 (0.002)
-0.001 (0.001)
0.001 (0.001)
2.019 (0.010)
2.017 (0.009)
2.017 (0.006)
2.018 (0.005)
0.689 (0.091)
0.498 (0.132)
0.777 (0.076)
0.786 (0.083)
0.596,0.461
0.591,0.456
0.547,0.408
0.593,0.472
0.980
0.980
0.020
0.980
JP
Maximizing Strategies (p )
• estimates of β show these two intervals are almost equally attractive for commercial absent any incentive to coordinate (e.g., 4-5pm if (β = 0.001, α = 0)
then p∗ = 0.5003)
• estimated α implies modest Nash equilibrium coordination on timing but joint
payoff maximization implies almost perfect coordination because externality
internalized
(iii) Testing for Multiple Equilibria Separately in Large Markets and Medium/Small
Markets
• if proportional (αP−i) formulation is incorrect then degree of coordination may
vary with market size/number of stations - different stories possible
Afternoon Drivetime hours
Non-Drivetime hours
3-4pm
4-5pm
5-6pm
6-7pm
3-4am
12-1pm
9-10pm
10-11pm
LRTS
0.3
0.1
0
0
0
0
4.2
0.0
Significant at
39%
38%
-
-
-
-
7.6%
43.6%
18.2
22.7
18.2
13.1
0.3
0
0
0
<0.2%
<0.2%
<0.2%
0.2%
31.6%
-
-
-
Markets Rank 1-29
Markets Ranks 30+
LRTS
Significant at
• also find slightly more equilibrium coordination in the medium and smaller
markets than all markets together (e.g., 4-5pm p∗A, p∗B is (0.610,0.449) cf.
(0.591,0.456) in all markets)
Extension/Robustness Check 1: “Clock Strategies”
• treating observations from same stations on different days as independent may
overstate significance
• look for coordination in scheduled (or average) timing
• assume clock strategy chosen periodically (monthly or annually) and then implemented with noise; stations benefit from coordination in actual timing, but
(β, ε) affect clock preferences
• look for multiple equilibria in choice of clock strategies
• find: (i) multiple equilibria in clock strategies during drivetime, (ii) more coordination on clock strategies than actual timing; (iii) almost perfect coordination
if joint payoff maximization or eliminate noisy implementation
(i) Evidence of Multiple Equilibria in Clock Strategy Choices (LRTS)
Afternoon drivetime
Non-drivetime hours
3-4pm
4-5pm
5-6pm
6-7pm
3-4am
12-1pm
9-10pm
10-11pm
21.7
25.3
15.9
15.1
0.1
0.0
1.4
0.0
<0.2%
<0.2%
<0.2%
<0.2%
44%
37%
11.2%
26%
7.5
16.6
7.1
9.9
3.0
0.3
0
0
<0.2%
<0.2%
0.4%
<0.2%
4%
24.8%
-
-
Monthly choice
LRTS
Significant at
Annual choice
LRTS
Significant at
• effect of fewer observations on clock strategies offset by more coordination in
clock strategies and better identification of strategies with more time series
observations
0
Kernel Density
1
2
3
Fit of 1 Eqm Model: Weekly Clock Strategy Model 12-1pm
0
.2
.4
.6
.8
1
Estimated Proportion of Stations in Market with Interval 1 for Clock Strategy
Actual Markets
Simulated Mkts +1 SD
Simulated Markets 1 Eqm
Simulated Mkts -1 SD
0
Kernel Density
1
2
3
Fit of 1 Eqm Model: Weekly Clock Strategy Model 4-5pm
0
.2
.4
.6
.8
1
Estimated Proportion of Stations in Market with Interval 1 for Clock Strategy
Actual Markets
Simulated Mkts +1 SD
Simulated Markets 1 Eqm
Simulated Mkts -1 SD
0
Kernel Density
1
2
3
Fit of 2 Eqa Model: Weekly Clock Strategy Model 4-5pm
0
.2
.4
.6
.8
1
Estimated Proportion of Stations in Market with Interval 1 for Clock Strategy
Actual Markets
Simulated Mkts +1 SD
Simulated Markets 2 Eqa
Simulated Mkts -1 SD
0
Kernel Density
1
2
3
Fit of 2 Eqa Model: Weekly Clock Strategy Model 5-6pm
0
.2
.4
.6
.8
1
Estimated Proportion of Stations in Market with Interval 1 for Clock Strategy
Actual Markets
Simulated Mkts +1 SD
Simulated Markets 2 Eqa
Simulated Mkts -1 SD
(ii) Estimates of (β, α, λ, q) and Strategies for Annual Choice Model 4-6pm
β
α
λ
q
Nash equilibrium clock
strategies (p*A,p*B)
Equilibrium strategies if
q=1
Joint expected payoff
maximizing strategies (pJP)
Number of market years
Number of station years
4-5pm
0.011 (0.028)
8.223 (0.476)
0.544 (0.167)
0.752 (0.006)
(0.695,0.400)
5-6pm
0.006 (0.004)
8.415 (0.615)
0.502 (0.124)
0.747 (0.006)
(0.660,0.418)
(0.999,0.001)
(0.999,0.001)
0.983
0.981
145
1,006
145
1,009
• q reflects how often scheduled interval chosen; lower q reduces incentive to
coordinate on clock strategies - if q = 1 equilibrium coordination on clock
strategies would be almost perfect
Extension/Robustness Check 2:Market Classification
• classify markets into different equilibria by calculating conditional probability
that observation m comes from equilibrium A
b
λ
¡
¢nm1 ¡
p∗A (b
β, b
α)
¢nm2
1 − p∗A (b
β, b
α)
A
b
τm = ¡
¢nm1 ¡
¢nm2
¡
¢nm1 ¡
¢nm2
b
λ p∗A (b
β, b
α)
1 − p∗A (b
β, b
α)
+ (1 − b
λ) p∗B (b
β, b
α)
1 − p∗B (b
β, b
α)
(15)
and, if all misclassification are equally costly, the Bayes Rule for classification
is
classify as equilibrium A ←→ τb A
m > 0.5
(16)
• two features of interest:
(a) medium-sized markets classified with greater confidence (even though less stations than the largest markets)
(b) stations in a small market where listeners tune in to stations in a larger market
(e.g., Worcester, MA and Boston, MA) tend to choose same times for commercials
(markets in same equilibrium)
Figure 4(a): Classification of US Metro-Markets into Two Equilibria
Based on the Results of the Annual Version of Model 2 for 5-6pm
Legend
Equilibrium A with posterior
probability > 0.75
Equilibrium A with posterior
probability between 0.5 and
0.75
Equilibrium B with posterior
probability between 0.5 and
0.75
Equilibrium B with posterior
probability > 0.75
Honolulu
(b) North East US
119
8
61
118
79
82
35
49
104
59
1
67
18
69
51
113
Legend
78
6
75
107
135
Equilibrium A with posterior
probability > 0.75
Equilibrium A with posterior
probability between 0.5 and
0.75
20
7
Equilibrium B with posterior
probability between 0.5 and
0.75
Equilibrium B with posterior
probability > 0.75
56
38
Market numbers are Arbitron metromarket ranks: see next page for key
Extension/Robustness Check 3: Market Characteristics Affect α and
λ
• parameterize α (incentive to coordinate) and λ (equilibrium selection) to depend on two characteristics: market size/number of stations and ownership
concentration
• less incentive to coordinate in the largest markets; but little evidence that
greater ownership concentration is associated with greater coordination (externality should be internalized)
• no systematic effects of these characteristics on equilibrium selection (not surprising in this context)
Extension/Robustness Check 4: Richer Analysis of Clock Strategies
• stations which do not usually have a break in either of these intervals are
nevertheless more likely to choose the coordination interval if they do happen
to have a break
• but some evidence that some stations who adopt the non-coordination interval
may be using counter-programming strategies (choose close but not identical
times for commercials)
6
Conclusions
1. this paper has shown how multiple equilibria, in the model and in the data, can
help to identify the effect of interactions between agents;
• the intuition for why multiple equilibria can help should generalize to more
complicated settings, but need to determine how to identify the mixture of
different equilibria in the data
• explicit modelling of multiple equilibria and equilibrium selection is important if
the data may contain multiple equilibria especially if we need to predict whether
the equilibrium a market is in may change
2. provided evidence that radio stations have an incentive to coordinate on the
timing of commercial breaks during drivetime, especially outside the largest
markets. The modest effect this incentive has on equilibrium strategies is not
inconsistent with listener avoidance of commercials having a large effect on the
value of radio advertising and industry revenues
Reduced Form Analysis of the Degree of Coordination
• model of coordination can predict more coordination in equilibrium in markets
with fewer stations, more concentrated ownership and more asymmetric listening (a leader can help in coordination); model of differentiation gives different
predictions
• form a measure of overlap of commercials which controls for aggregate withinhour patterns, regress on market characteristics
• find, especially during drivetime and to a lesser extent midday hours, more
coordination in markets with (i) fewer stations, (ii) less listening to outside of
market stations, (iii) more concentrated ownership and (iv) more asymmetric
listenership and, in particular, find stations coordinate with the largest station
in asymmetric markets; mixed effects of quantity of commercials.