lower bound on profits p H p L p L p H p H 2 , 2 0 , 3 p L p H p L 3 , 0 1 , 1 • Suppose firm 1 uses “tit for tat” and firm 2 has a revision on the first day – 2 ’s profits from choosing p L on day 1 are bounded above by v L ˆ 2 = 1 + (1 − µ )1 + µ 3 = 2 + 2 µ ���� ���� � �� � day 1 day 2 day 2 1 revises 1 doesn’t revise 11 / 33
lower bound on profits p H p L p L p H p H 2 , 2 0 , 3 p L p H p L 3 , 0 1 , 1 • Suppose firm 1 uses “tit for tat” and firm 2 has a revision on the first day – 2 ’s profits from choosing p L on day 1 are bounded above by v L ˆ 2 = 1 + (1 − µ )1 + µ 3 = 2 + 2 µ – 2 ’s profits from choosing p H on day 1 and p L on day 2 are bounded below by v H = 0 + (1 − µ )3 + µ 1 = 3 − 2 µ ���� ���� � �� � day 1 day 2 day 2 1 revises 1 doesn’t revise 11 / 33
lower bound on profits p H p L p L p H p H 2 , 2 0 , 3 p L p H p L 3 , 0 1 , 1 • Suppose firm 1 uses “tit for tat” and firm 2 has a revision on the first day – 2 ’s profits from choosing p L on day 1 are bounded above by v L ˆ 2 = 1 + (1 − µ )1 + µ 3 = 2 + 2 µ – 2 ’s profits from choosing p H on day 1 and p L on day 2 are bounded below by v H 2 = 0 + (1 − µ )3 + µ 1 = 3 − 2 µ – If µ < 1 / 4 then v H v L 2 > ˆ 2 11 / 33
lower bound on profits p H p L p L p H p H 2 , 2 0 , 3 p L p H p L 3 , 0 1 , 1 • Suppose firm 1 uses “tit for tat” and firm 2 has a revision on the first day – If µ < 1 / 4 then firm 2 chooses p H on day 1 • If µ < 1 / 4 , firm 1 can guarantee profits above 2 by using “tit for tat” – If firm 2 sets p L on both days, firm 1 makes 2 in profits – If firm 2 sets p M on at least one day, firm 1 makes at least 3 in profits – If firm 2 has a revision on day 1 it sets p H 12 / 33
lower bound on profits p H p L p L p H p H 2 , 2 0 , 3 p L p H p L 3 , 0 1 , 1 • Suppose firm 1 uses “tit for tat” and firm 2 has a revision on the first day – If µ < 1 / 4 then firm 2 chooses p H on day 1 • If µ < 1 / 4 , firm 1 can guarantee profits above 2 by using “tit for tat” If revisions are sufficiently unlikely, joint profits in any subgame- perfect equilibrium are strictly greater than 4 12 / 33
word 1. introduction 2. example 3. model 4. main result 5. closing remarks
• Two symmetric firms j ∈ { 1 , 2 } • Continuous time t ∈ [0 , ∞ ) • Consumers arrive randomly – Poisson process with parameter λ > 0 – ( y n ) denotes sequence of arrival times – A single consumer arrives at each y n 13 / 33
stage game P = R + 14 / 33
stage game π j : P 2 → R + P = R + 14 / 33
stage game � π j : P 2 → R + � � p ∈ P 2 � P = R + Π = π ( p ) π 2 π M b π ( p ) π π 1 14 / 33
pricing algorithms • Pricing algorithms set current prices contingent on the history of past prices • Finite automata a = (Ω , ω 0 , θ, α ) – Finite set of states Ω – Initial state ω 0 – Pricing rule α : Ω → P – Measurable transition function θ : Ω × P → Ω p − j else ∗ ∗ ∗ p j p M 0 p M p M 0 ∗ ∗ always monopolistic grim trigger two monopolistic 15 / 33
dynamic game a 0 a ′ a ′′ 2 2 2 a 0 a ′ a ′′ a ′′′ 1 1 1 1 • Firms simultaneously set algorithms at time t = 0 and can revise them at exogenous stochastic times – Poisson process with parameter µ > 0 – Arrival of revision is independent across firms and independent of consumer-arrival times 16 / 33
dynamic game a 0 a ′ a ′′ 2 2 2 a 0 a ′ a ′′ a ′′′ 1 1 1 1 • Firms simultaneously set algorithms at time t = 0 and can revise them at exogenous stochastic times • A strategy s j : H j → ∆( A ) for firm j chooses algorithms – As a function of past algorithms, prices, and number of past consumers 16 / 33
dynamic game a 0 a ′ a ′′ 2 2 2 a 0 a ′ a ′′ a ′′′ 1 1 1 1 • Firms simultaneously set algorithms at time t = 0 and can revise them at exogenous stochastic times • A strategy s j : H j → ∆( A ) for firm j chooses algorithms – As a function of past algorithms, prices, and number of past consumers – In this talk, not as a function of clock time of consumer and revision arrivals 16 / 33
solution concept • Firms maximize (normalized) expected discounted profits � � ∞ r � v j = λ + r × E exp( − ry n ) π j ( p n ) n =1
solution concept • Firms maximize (normalized) expected discounted profits � � ∞ r � v j = λ + r × E exp( − ry n ) π j ( p n ) n =1 ∞ r � = λ + r × E [ exp( − ry n ) ] E [ π j ( p n ) ] n =1 17 / 33
solution concept • Firms maximize (normalized) expected discounted profits � � ∞ r � v j = λ + r × E exp( − ry n ) π j ( p n ) n =1 ∞ r � = λ + r × E [ exp( − ry n ) ] E [ π j ( p n ) ] n =1 � � n ∞ r λ � = λ + r × E [ π j ( p n ) ] λ + r n =1 17 / 33
solution concept • Firms maximize (normalized) expected discounted profits � � n ∞ λ r � v j = λ + r × E [ π j ( p n ) ] λ + r n =1 17 / 33
solution concept • Firms maximize (normalized) expected discounted profits � � n ∞ λ r � v j = λ + r × E [ π j ( p n ) ] λ + r n =1 • Sub-game perfect Nash equilibria s ∈ S ∗ 17 / 33
solution concept • Firms maximize (normalized) expected discounted profits � � n ∞ λ r � v j = λ + r × E [ π j ( p n ) ] λ + r n =1 • Sub-game perfect Nash equilibria s ∈ S ∗ • Using Levy (2015) and Mertens and Parthasarathy (1987) If the profit function π is bounded (and Borel measurable), then the dynamic game has an equilibrium 17 / 33
word 1. introduction 2. example 3. model 4. main result 5. closing remarks
inevitability of collusion • Fix any interest rate r and any constant ε > 0 18 / 33
inevitability of collusion • Fix any interest rate r and any constant ε > 0 • Let t 0 be the (random) first date at which each of the two firms has had at least one revision opportunity 18 / 33
inevitability of collusion • Fix any interest rate r and any constant ε > 0 • Let t 0 be the (random) first date at which each of the two firms has had at least one revision opportunity • If costumers arrive frequently λ > r λ • And revisions are infrequent 0 < µ < r ¯ µ ( ε, λ ) 18 / 33
inevitability of collusion • Fix any interest rate r and any constant ε > 0 • Let t 0 be the (random) first date at which each of the two firms has had at least one revision opportunity • If costumers arrive frequently λ > r λ • And revisions are infrequent 0 < µ < r ¯ µ ( ε, λ ) • For any date τ ≥ t 0 the joint continuation profits are closer than ε from the joint monopolistic profits with probability greater than (1 − ε ) in any equilibrium, i.e. � � π M − ε s ∈ S ∗ Pr s inf v τ > ¯ ¯ > 1 − ε 18 / 33
inevitability of collusion π 2 π M π π 1 19 / 33
inevitability of collusion π 2 π M π π 1 19 / 33
step 1 π 2 π M π π 1 � Π j = � � a − j ∈ BR( a j ) � v ( a ) 20 / 33
step 1 π 2 π M π π 1 If λ r > λ , then Π j intersects the Pareto frontier 21 / 33
step 2 • Suppose current algorithms induce a sequence of profits π n • Expected discounted profits can be decomposed as � r j + w 0 � λπ 1 + 1 1 · w 1 j + 1 2 · w 2 v j = E exp( − rz 1 ) 1 0 · j � �� � � �� � � �� � discounting revisions consumer to first event
step 2 • Suppose current algorithms induce a sequence of profits π n • Expected discounted profits can be decomposed as � r j + w 0 � λπ 1 + 1 1 · w 1 j + 1 2 · w 2 v j = E exp( − rz 1 ) 1 0 · j � �� � � �� � � �� � discounting revisions consumer to first event � r � λπ 1 j + w 0 + Pr(1) w 1 j + Pr(2) w 2 = E [ exp( − rz 1 ) ] Pr(0) j j � �� � � �� � � �� � discounting revsion consumer 22 / 33
step 2 • Suppose current algorithms induce a sequence of profits π n • Expected discounted profits can be decomposed as � r j + w 0 � λπ 1 + 1 1 · w 1 j + 1 2 · w 2 v j = E exp( − rz 1 ) 1 0 · j � �� � � �� � � �� � discounting revisions consumer to first event � r � λπ 1 j + w 0 + Pr(1) w 1 j + Pr(2) w 2 = E [ exp( − rz 1 ) ] Pr(0) j j � �� � � �� � � �� � discounting revsion consumer r λ µ µ r + λ + 2 µπ 1 r + λ + 2 µ w 0 r + λ + 2 µ w 1 r + λ + 2 µ w 2 = j + + j + j j � �� � � �� � consumer revision 22 / 33
step 2 • Suppose current algorithms induce a sequence of profits π n • Expected discounted profits can be decomposed as r λ µ µ r + λ + 2 µπ 1 r + λ + 2 µ w 0 r + λ + 2 µ w 1 r + λ + 2 µ w 2 v j = j + j + j + j 22 / 33
step 2 • Suppose current algorithms induce a sequence of profits π n • Expected discounted profits can be decomposed as r λ µ µ r + λ + 2 µπ 1 r + λ + 2 µ w 0 r + λ + 2 µ w 1 r + λ + 2 µ w 2 v j = j + j + j + j • Iterating this process yields ∞ r 2 µ � β k π k v j = r + 2 µ (1 − β ) j + r + 2 µ ˜ w j k =0 where β = λ/ ( r + λ + 2 µ ) 22 / 33
step 2 p 0 π 2 − j else p 0 π M j 0 b π ( p 0 ) ∗ grim-trigger algorithm a 0 j π π 1 In any equilibrium,if firm − j observes a 0 j , it chooses an algorithm − j for at least N = c 1 ( p 0 ) r that mimics a 0 µ − c 0 consumers 23 / 33
b step 2 p 0 π 2 − j else p 0 π M j 0 b π ( p 0 ) ∗ grim-trigger algorithm a 0 j π π 1 If µ µ ( ε, λ ) , then continuation values at the moment of each revision r < ¯ after the first one are close to the Pareto frontier of Π j 24 / 33
step 3 ¯ v t π M t Revision continuation joint profits after t 0 are close enough to π M so that, after the second revision, long run profits remain high 25 / 33
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