A Game Theoretic Model for Network
pg
Decisions
Upgrade
Sept 27, 2006
Allerton Conference
John Musacchio
Assistant Professor
Technology
gy and Information Management
g
University of California, Santa Cruz
[email protected]
Joint work with:
Jean Walrand
UC Berkeley
Shuang Wu
UC Santa Cruz
Starting Network
$
Provider 1
$
Provider 2
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Network After One Upgrade
Both Provider
Provider’s
s Revenues Increase
a
f
$
$
$
$
Users less
dissatisfied
Provider 1
Provider 2
Provider 1 invests U to upgrade.
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Network After Both Upgrade
Both Provider’s Revenues Increase
R
$
$
R
$
$
Users more
satisfied
Provider 1
Provider 2
Provider 2 invests $U to upgrade
upgrade.
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Outline
z
Two Providers
z
N providers
p
z
Upgrade costs that decline
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Multi Stage Game
Multi-Stage
z
Payoffs n periods in future discounted by δn.
z
In each period, players simultaneously decide
whether to upgrade.
z
A player sees what the other player did in previous
p
periods.
z
A strategy si is a plan of action for each period k as a
function of the history hk.
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Per-Period Revenue Matrix:
Not Upgrade
Upgrade
Not Upgrade
0,0
f, a
Upgrade
a, f
R, R
• The provider incurs a one time cost of
U when upgrading.
• Upgrading is irreversible.
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Solution Concepts
z
An ordinary Subgame Perfect Nash Equilibrium
(SPE) is
–
a strategy profile
z
–
–
z
(
(assignment
i
t off strategies
t t i to
t allll players)
l
)
such that in any subgame,
no player can improve her payoff by unilaterally
deviating from the strategy profile.
SPE Condition:
- For all
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SPE by Iterated Strict dominance
z Strategy si is dominated if
For some j and all s-i
Idea of Iterated Strict Dominance:
z Make G’ by removing dominated strategies from
original game G.
z Repeat…
z
If after
f this
hi process only
l one strategy profile
fil remains,
i
we say it is an SPE by iterated strict dominance.
–
Players can deduce what opponents will do, without
“guessing.”
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Theorem 1
R R R R
Suppose
…
Equivalently:
>
f f f f
…
U
Then an SPE is:
z Upgrade Immediately – Both providers choose to upgrade in the
current period
If Also
Then
z
Upgrade Immediately is a unique SPE by iterated
strict dominance
The following are also SPE:
z No First Upgrade – No provider willing to upgrade first
z
Delayed Upgrade – Each provider waits until period n
to upgrade
z
Mixed Upgrade – Each provider upgrades with
probability α in each period.
Theorem 1 Continued
Equivalently:
Suppose:
R R R R
…
<
f f f f
…
U
If Also
Then an SPE iis:
Th
z Never upgrade – Do not upgrade no matter what
z
Asymmetric Freeride – One player upgrades the
other freerides
z
Mixed Freeride – Players upgrade with
probability α, until one upgrades. Then other
one freerides
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Th
Theorem
1
z
Two Groupings of Cases:
Group 1:
R
R
R
R
…
>
f
f
f
f
f
f
f
f
…
U
Group 2:
R
R
R
R
…
<
…
U
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“Upgrade Immediately”
z
Both providers Upgrade in current period.
SPE if
z
“Proof”
z
Stick
Deviate
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Upgrade Immediately (Cond. For Uniqueness)
Suppose:
Upgrade
Not
A upgrades
z
z
Slot i
Slot i+1
If A upgrades
d in
i slot
l t ii, B upgrades
d b
by ii+1
1
Don’t know what B does in slot i
B
Upgrade i
upgrade
A
Wait
Wait
U
Upgrade
d IImmediately
di t l (Uniqueness)
(U i
)
z
Upgrade is better for both conditional payoff
functions if
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Delayed
y Upgrade
pg
z
z
Each provider expects the other to upgrade
in period n.
Srtategy
–
–
z
z
Upgrade in period n.
But,, If opponent
pp
upgrades
pg
in p
period i<n,, upgrade
pg
in i+1.
n=∞ corresponds to “no first upgrade.”
Conditions:
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Delayed Upgrade Proof
z
Suppose player B plays “upgrade in period n”
Upgrade at 0
Upgrade at n
A
Freeride
z
Upgrade at n best if:
“Mixed Upgrade”
z
Subgame starting with no-one upgraded:
–
z
Subgame starting with one-upgraded:
–
z
Each upgrades with probability α.
The not-upgraded upgrades immediately.
SPE if
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Mixed Upgrade
z
Suppose B follows mixed upgrade and that
Upgrade
Wait
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Mixed Upgrade
z
Difference
z
Sufficient condition for existence of a zero
Th
Theorem
1 prooff
z
Two Groupings
p g of Cases:
Group 1:
R
R
R
R
…
>
f
f
f
f
f
f
f
f
…
U
Group 2:
R
U
R
R
R
…
<
…
Asymmetric Free-ride
Free ride
z
Player A upgrades
Pl
d slot
l t1
1.
Player B never upgrades.
z
SPE if
z
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Other Cases for
z
“Mixed Freeride”
–
–
–
z
Condition:
Each upgrades with probability α; if one upgrades
first the other freerides
freerides.
Proof: Similar to “Mixed Upgrade.”
“Never
Never Upgrade
Upgrade”
–
–
–
Condition:
No one upgrades,
pg
even if the other one were to.
Proof: easy.
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Outline
z
Two Providers
z
N providers
p
z
Upgrade costs that decline
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N-Providers in Tandem
z
Let
–
f(j) = free-rider benefit if j providers upgrade.
z
–
a(j) = early adopter benefit if j providers upgrade.
z
z
f(0) == 0
a(N) == R
All U
Upgrade
d a SPE if:
if
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U
Upgrade
d IImmediately
di t l (Dominance)
(D
i
)
z
Suppose we enter a subgame with 2 not-upgraded
upgrade
freeride
upgrade
freeride
z
Upgrading Dominant if:
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Upgrade Immediately (Dominance)
z
Induction argument shows that everyone
upgrading
di iis unique
i
SPE b
by it
iterated
t dd
deletion
l ti
of dominated strategies if:
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Outline
z
Two Providers
z
N providers
p
z
Upgrade costs that decline
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Declining Upgrade Cost Model
z
z
Continuous Time
F t
Future
Revenues
R
are discounted
di
t d att rate
t δ
–
z
Example if both providers upgrade at time 0, the
P V of each
P.V.
each’s
s revenues is:
Upgrade costs decline at rate γ
–
–
γ includes declining costs + discounting so γ>δ.
γ>δ
P.V. of upgrade cost at time t’ is:
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Critical Times
z
Time when upgrading looks better than
freeriding:
z
Time when being a first adopter looks
attractive:
Theorem 3
Suppose ta* ≥ tf*
z Then the only SPE is:
–
z
Both providers upgrade at time tf*
Proof outline:
–
Step
p 1: A upgrades
pg
before tf*
Æ B upgrades at tf* (dominant)
–
Step 2: A upgrades after tf* and B not yet upgraded
Æ B Upgrades immediately after A (dominant).
–
Step 3: Because A can induce B to upgrade at tf*, A
best option is to upgrade at tf*
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Theorem 3 Proof – Step 1
z
z
Suppose A upgrades at time t’
t < tf*
B’s best response in ensuing subgame:
Rev. after upgrade
Freeride revenue before upgrade
Upgrade cost
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Th
Theorem
3 Proof
P
f – Step
St 2
z
z
Suppose A upgrades at time t’
t > tf*
B’s best response in ensuing subgame:
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Theorem 3 Proof – Step 3
z
z
z
Suppose A considers
S
id
upgrading
di b
before
f
tf*.
Considering induced behavior on B, A’s best
response is:
Similarly, can show that A upgrading after tf*
gives lower payoff.
Theorem 4
z
z
Suppose ta* < tf*
S
It is a SPE for
–
–
z
One provider
O
id tto upgrade
d att titime ta*
The other provider to upgrade at time tf*
Proof:
–
Similar as Theorem 3’s proof.
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Time to Upgrade,
pg
, δ = 0.05
14
Time tto Upgra
ade
12
10
8
6
4
γ =0.20
2
0
0
2
4
6
γ =0.15
8
γ =0.10
10 12
14
16 18 20
Conclusions
z
Discrete Time Model
–
Strong conditions required to have unique SPE.
z
–
z
Otherwise many strange SPE possible.
Freeriding effect may prevent upgrades even
when it is socially optimal to do so.
Continuous Time with Declining Costs
–
–
–
–
Unique SPE if τf* ≤ τa*
Asymmetric SPE otherwise.
Freeriding delays upgrades.
More rapidly declining upgrade costs may
increase the time until networks upgrade.
upgrade
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