Redistribution through Markets Our Model good K (indivisible) and money M (divisible). Unit mass of sellers (own good K ). Mass µ of buyers (don’t yet own good K ). Each agent has a valuation v K for K . If ( x K , x M ) denotes the holdings of K and M , then utility ∝ v K · x K + x M . Dworczak � Kominers R � Akbarpour R April 20, 2020 13
Redistribution through Markets Our Model good K (indivisible) and money M (divisible). Unit mass of sellers (own good K ). Mass µ of buyers (don’t yet own good K ). Each agent has a valuation v K for K . Each agent has a valuation v M for M . If ( x K , x M ) denotes the holdings of K and M , then utility ∝ v K · x K + v M · x M . Dworczak � Kominers R � Akbarpour R April 20, 2020 13
Redistribution through Markets Our Model good K (indivisible) and money M (divisible). Unit mass of sellers (own good K ). Mass µ of buyers (don’t yet own good K ). Each agent has a valuation v K for K . Each agent has a valuation v M for M . If ( x K , x M ) denotes the holdings of K and M , then utility ∝ v K · x K + v M · x M . Dworczak � Kominers R � Akbarpour R April 20, 2020 13
Redistribution through Markets Our Model good K (indivisible) and money M (divisible). Unit mass of sellers (own good K ). Mass µ of buyers (don’t yet own good K ). Each agent has a valuation v K for K . Each agent has a valuation v M for M . If ( x K , x M ) denotes the holdings of K and M , then utility ∝ v K · x K + v M · x M . ❀ Rates of substitution v K v M =: r ∼ G j ( r ) ( ∼ Unif [0 , 1] for now). Dworczak � Kominers R � Akbarpour R April 20, 2020 13
Redistribution through Markets Simple Mechanisms under Uniform Distribution First, we solve one-sided problems given Q and R . Then, we link our characterizations of seller- and buyer-side solutions through the optimal choice of Q and R . Our general results have the same structure/intuition(!). Dworczak � Kominers R � Akbarpour R April 20, 2020 14
Redistribution through Markets Simple Mechanisms under Uniform Distribution First, we solve one-sided problems given Q and R . Then, we link our characterizations of seller- and buyer-side solutions through the optimal choice of Q and R . Our general results have the same structure/intuition(!). Key Observation Agents’ behavior depends only on r = v K v M , but r is informative about welfare weight λ ( r ) := E [ v M | v K v M = r ] . Dworczak � Kominers R � Akbarpour R April 20, 2020 14
Redistribution through Markets Simple Mechanisms under Uniform Distribution First, we solve one-sided problems given Q and R . Then, we link our characterizations of seller- and buyer-side solutions through the optimal choice of Q and R . Our general results have the same structure/intuition(!). Key Observation Agents’ behavior depends only on r = v K v M , but r is informative about welfare weight λ ( r ) := E [ v M | v K v M = r ] . ⇒ Identifying “poorer” agents through market behavior. Dworczak � Kominers R � Akbarpour R April 20, 2020 14
Redistribution through Markets Measures of Inequality Λ j := E j [ v M ] Dworczak � Kominers R � Akbarpour R April 20, 2020 15
Redistribution through Markets Measures of Inequality Λ j := E j [ v M ] cross-side inequality if Λ S � = Λ B Dworczak � Kominers R � Akbarpour R April 20, 2020 15
Redistribution through Markets Measures of Inequality Λ j := E j [ v M ] cross-side inequality if Λ S � = Λ B same-side inequality on side j ∈ { B , S } if λ j �≡ Λ j low if λ j ( r j ) ≤ 2Λ j ; high otherwise Dworczak � Kominers R � Akbarpour R April 20, 2020 15
Redistribution through Markets The Optimal Seller Price p S Goal: Acquire Q objects while spending at most R . � p S � � Q max λ S ( r )( p S − r ) dG S ( r ) + Λ S ( R − p S Q ) . G S ( p S ) p S ≥ G − 1 ( Q ) r S S Dworczak � Kominers R � Akbarpour R April 20, 2020 16
Redistribution through Markets The Optimal Seller Price p S Goal: Acquire Q objects while spending at most R . � p S � � Q max λ S ( r )( p S − r ) dG S ( r ) + Λ S ( R − p S Q ) . G S ( p S ) p S ≥ G − 1 ( Q ) r S S Three effects of pushing p S above p C S : 1 allocative efficiency ↓ ; 2 lump sum transfer R − p S Q ↓ ; 3 money to sellers who trade ↑ . Dworczak � Kominers R � Akbarpour R April 20, 2020 16
Redistribution through Markets The Optimal Seller Price p S Goal: Acquire Q objects while spending at most R . � p S � � Q max λ S ( r )( p S − r ) dG S ( r ) + Λ S ( R − p S Q ) . G S ( p S ) p S ≥ G − 1 ( Q ) r S S Proposition When seller same-side inequality is low, p S = p C S is optimal. When seller same-side inequality is high and Q is low enough, rationing at a price p S > p C S is optimal. Dworczak � Kominers R � Akbarpour R April 20, 2020 16
Redistribution through Markets The Optimal Seller Price p S Goal: Acquire Q objects while spending at most R . � p S � � Q max λ S ( r )( p S − r ) dG S ( r ) + Λ S ( R − p S Q ) . G S ( p S ) p S ≥ G − 1 ( Q ) r S S Proposition When seller same-side inequality is low, p S = p C S is optimal. When seller same-side inequality is high and Q is low enough, rationing at a price p S > p C S is optimal. Decision to trade always identifies sellers with low rates of substitution ( ⇒ equity ↑ )! Dworczak � Kominers R � Akbarpour R April 20, 2020 16
Redistribution through Markets The Optimal Buyer Price p B Goal: Allocate Q objects with revenue at least R . � ¯ r B � Q � max λ B ( r )( r − p B ) dG B ( r ) + Λ B ( p B Q − R ) . 1 − G B ( p B ) p B ≤ G − 1 B (1 − Q ) p B Dworczak � Kominers R � Akbarpour R April 20, 2020 17
Redistribution through Markets The Optimal Buyer Price p B Goal: Allocate Q objects with revenue at least R . � ¯ r B � Q � max λ B ( r )( r − p B ) dG B ( r ) + Λ B ( p B Q − R ) . 1 − G B ( p B ) p B ≤ G − 1 B (1 − Q ) p B Three effects of pushing p B below p C B : 1 allocative efficiency ↓ ; 2 lump sum transfer p B Q − R ↓ ; 3 “money to” buyers who buy ↑ . Dworczak � Kominers R � Akbarpour R April 20, 2020 17
Redistribution through Markets The Optimal Buyer Price p B Goal: Allocate Q objects with revenue at least R . � ¯ r B � Q � max λ B ( r )( r − p B ) dG B ( r ) + Λ B ( p B Q − R ) . 1 − G B ( p B ) p B ≤ G − 1 B (1 − Q ) p B Proposition Setting p B = p C B is optimal (“fullstop”). Dworczak � Kominers R � Akbarpour R April 20, 2020 17
Redistribution through Markets The Optimal Buyer Price p B Goal: Allocate Q objects with revenue at least R . � ¯ r B � Q � max λ B ( r )( r − p B ) dG B ( r ) + Λ B ( p B Q − R ) . 1 − G B ( p B ) p B ≤ G − 1 B (1 − Q ) p B Proposition Setting p B = p C B is optimal (“fullstop”). Decision to trade always identifies buyers with higher rates of substitution ( ⇒ equity ↓ )! Dworczak � Kominers R � Akbarpour R April 20, 2020 17
Redistribution through Markets Multiple Prices? Single price p S is optimal for seller-side. Dworczak � Kominers R � Akbarpour R April 20, 2020 18
Redistribution through Markets Multiple Prices? Single price p S is optimal for seller-side. Not so for buyers! Dworczak � Kominers R � Akbarpour R April 20, 2020 18
Redistribution through Markets Multiple Prices? Single price p S is optimal for seller-side. Not so for buyers! Using two prices can improve screening: buy at p H B with probability 1 or at p L B with probability δ < 1 ❀ “wealthier” buyers choose p H B . Dworczak � Kominers R � Akbarpour R April 20, 2020 18
Redistribution through Markets Multiple Prices? Single price p S is optimal for seller-side. Not so for buyers! Using two prices can improve screening: buy at p H B with probability 1 or at p L B with probability δ < 1 ❀ “wealthier” buyers choose p H B . Proposition When buyer same-side inequality is low, p H B = p C B is optimal. When buyer same-side inequality is high and Q is large enough, rationing at p L B is optimal. Dworczak � Kominers R � Akbarpour R April 20, 2020 18
Redistribution through Markets Multiple Prices? Single price p S is (globally) optimal for seller-side. Not so for buyers! Using two prices can improve screening: buy at p H B with probability 1 or at p L B with probability δ < 1 ❀ “wealthier” buyers choose p H B . Proposition When buyer same-side inequality is low, p H B = p C B is (globally) optimal. When buyer same-side inequality is high and Q is large enough, rationing at p L B is (globally) optimal. Dworczak � Kominers R � Akbarpour R April 20, 2020 18
Redistribution through Markets Linking the Two Sides (I) Proposition When seller same-side inequality is low, p S = p C S is optimal. When seller same-side inequality is high and Q is low enough, rationing at a price p S > p C S is optimal. Proposition When buyer same-side inequality is low, p H B = p C B is optimal. When buyer same-side inequality is high and Q is large enough, rationing at p L B is optimal. Dworczak � Kominers R � Akbarpour R April 20, 2020 19
Redistribution through Markets Linking the Two Sides (II) – price wedges Proposition When same-side inequality is low on both sides, it is optimal to set prices such that the market clears, G S ( p S ) = µ (1 − G B ( p B )) ; we redistribute any revenue as a lump-sum payment to the side with higher average value for money Λ j . Dworczak � Kominers R � Akbarpour R April 20, 2020 20
Redistribution through Markets Linking the Two Sides (III) – rationing Proposition If seller same-side inequality is high and Λ S ≥ Λ B , then (if µ is low enough) it is optimal to ration the sellers by setting a single price above the market-clearing level. Dworczak � Kominers R � Akbarpour R April 20, 2020 21
Redistribution through Markets Linking the Two Sides (III) – rationing Proposition If seller same-side inequality is high and Λ S ≥ Λ B , then (if µ is low enough) it is optimal to ration the sellers by setting a single price above the market-clearing level. Proposition If buyer same-side inequality is high and buyers’ willingness to pay is sufficiently high [relative to the seller’s rates of substitution], then it is optimal to ration the buyers for µ ∈ (1 , 1 + ǫ ) . Dworczak � Kominers R � Akbarpour R April 20, 2020 21
Redistribution through Markets Today – (P)review How should we design marketplaces in the presence of systematic wealth inequality? ⋆ Model inequality as dispersion in marginal values for money. 1,2 ⇔ allowing for arbitrary Pareto weights in mechanism design. ❀ We seek to maximize weighted surplus subject to resource, budget balance, incentive, and individual rationality constraints. ⋆ Optimal Mechanism uses two instruments: cross-side inequality ⇒ price wedge ∗ same-side inequality ⇒ rationing 1 Following an approach from public finance. 2 Implicit assumption: this is a good approximation. Dworczak � Kominers R � Akbarpour R April 20, 2020 22
Redistribution through Markets Main Result For the full model (( v K , v M ) ∼ f { B , S } ), we seek to maximize welfare subject to incentive compatibility, individual rationality, market- clearing, and budget balance constraints. Dworczak � Kominers R � Akbarpour R April 20, 2020 23
Redistribution through Markets Main Result For the full model (( v K , v M ) ∼ f { B , S } ), we seek to maximize welfare subject to incentive compatibility, individual rationality, market- clearing, and budget balance constraints. Theorem There exists an optimal mechanism that uses some combination of a price wedge (with lump-sum transfer) and rationing. Dworczak � Kominers R � Akbarpour R April 20, 2020 23
Redistribution through Markets Main Result For the full model (( v K , v M ) ∼ f { B , S } ), we seek to maximize welfare subject to incentive compatibility, individual rationality, market- clearing, and budget balance constraints. Theorem There exists an optimal mechanism that uses some combination of a price wedge (with lump-sum transfer) and rationing. Total number of prices involved is small ( 4 at most; 3 if nonzero lump-sum transfer). Dworczak � Kominers R � Akbarpour R April 20, 2020 23
Redistribution through Markets Sketch of Proof 1 We transform the problem into maximization over allocation rules alone. Dworczak � Kominers R � Akbarpour R April 20, 2020 24
Redistribution through Markets Sketch of Proof 1 We transform the problem into maximization over allocation rules alone. 2 We represent a feasible allocation rule as a lottery over quantities ⇒ deterministic outcome in aggregate. Dworczak � Kominers R � Akbarpour R April 20, 2020 24
Redistribution through Markets Sketch of Proof 1 We transform the problem into maximization over allocation rules alone. 2 We represent a feasible allocation rule as a lottery over quantities ⇒ deterministic outcome in aggregate. 3 Market-clearing ⇔ expected quantity sold to buyers equals the expected quantity bought from sellers – essentially, a Bayes-plausibility constraint. Dworczak � Kominers R � Akbarpour R April 20, 2020 24
Redistribution through Markets Sketch of Proof 1 We transform the problem into maximization over allocation rules alone. 2 We represent a feasible allocation rule as a lottery over quantities ⇒ deterministic outcome in aggregate. 3 Market-clearing ⇔ expected quantity sold to buyers equals the expected quantity bought from sellers – essentially, a Bayes-plausibility constraint. 4 The solution is characterized as a concave closure of the objective function, subject to a linear constraint. Dworczak � Kominers R � Akbarpour R April 20, 2020 24
Redistribution through Markets Sketch of Proof 1 We transform the problem into maximization over allocation rules alone. 2 We represent a feasible allocation rule as a lottery over quantities ⇒ deterministic outcome in aggregate. 3 Market-clearing ⇔ expected quantity sold to buyers equals the expected quantity bought from sellers – essentially, a Bayes-plausibility constraint. 4 The solution is characterized as a concave closure of the objective function, subject to a linear constraint. 5 By Carath´ eodory’s Theorem, the solution is supported on at most three points ❀ three-price characterization for each side. (And two constraints are common across the market!) Dworczak � Kominers R � Akbarpour R April 20, 2020 24
Redistribution through Markets When to use each instrument? (modulo reg. conditions) Cross-side inequality (Λ B � = Λ S ) ❀ price wedge. ❀ moves money to the poorer side of the market. N.B. Drawback: Each seller receives the same lump-sum transfer regardless of how much she contributes to social welfare. Dworczak � Kominers R � Akbarpour R April 20, 2020 25
Redistribution through Markets When to use each instrument? (modulo reg. conditions) Cross-side inequality (Λ B � = Λ S ) ❀ price wedge. ❀ moves money to the poorer side of the market. N.B. Drawback: Each seller receives the same lump-sum transfer regardless of how much she contributes to social welfare. High seller-side inequality ❀ rationing. ❀ shifts more money to the sellers who are most eager to trade – identifies the “poorest.” Dworczak � Kominers R � Akbarpour R April 20, 2020 25
Redistribution through Markets When to use each instrument? (modulo reg. conditions) Cross-side inequality (Λ B � = Λ S ) ❀ price wedge. ❀ moves money to the poorer side of the market. N.B. Drawback: Each seller receives the same lump-sum transfer regardless of how much she contributes to social welfare. High seller-side inequality ❀ rationing. ❀ shifts more money to the sellers who are most eager to trade – identifies the “poorest.” High buyer-side inequality + large amount of trade ❀ rationing with multiple prices. N.B. The “poorest” buyers are those that are least able to trade! ❀ Rationing at a single price helps richer buyers more than poorer. Dworczak � Kominers R � Akbarpour R April 20, 2020 25
Redistribution through Markets Optimal Market Design under Wealth Inequality Dworczak � Kominers R � Akbarpour R April 20, 2020 26
Redistribution through Markets Optimal Market Design under Wealth Inequality Dworczak � Kominers R � Akbarpour R April 20, 2020 27
Redistribution through Markets Optimal Market Design under Wealth Inequality Dworczak � Kominers R � Akbarpour R April 20, 2020 28
Redistribution through Markets Optimal Market Design under Wealth Inequality Dworczak � Kominers R � Akbarpour R April 20, 2020 29
Redistribution through Markets Optimal Market Design under Wealth Inequality Dworczak � Kominers R � Akbarpour R April 20, 2020 30
Redistribution through Markets Optimal Market Design under Wealth Inequality Dworczak � Kominers R � Akbarpour R April 20, 2020 31
Redistribution through Markets Optimal Market Design under Wealth Inequality Dworczak � Kominers R � Akbarpour R April 20, 2020 32
Redistribution through Markets Optimal Market Design under Wealth Inequality Dworczak � Kominers R � Akbarpour R April 20, 2020 33
Redistribution through Markets Optimal Market Design under Wealth Inequality Dworczak � Kominers R � Akbarpour R April 20, 2020 34
Redistribution through Markets Optimal Market Design under Wealth Inequality Dworczak � Kominers R � Akbarpour R April 20, 2020 35
Redistribution through Markets Optimal Market Design under Wealth Inequality Dworczak � Kominers R � Akbarpour R April 20, 2020 36
Redistribution through Markets Coming Soon: New Paper! Dworczak � Kominers R � Akbarpour R April 20, 2020 37
Redistribution through Markets Coming Soon: New Paper! We introduce quality q ; rate of substitution r now characterizes trade-off between q and money. Agents also have a label representing observable characteristics (linked to r and/or Pareto weights). Dworczak � Kominers R � Akbarpour R April 20, 2020 38
Redistribution through Markets Coming Soon: New Paper! We introduce quality q ; rate of substitution r now characterizes trade-off between q and money. Agents also have a label representing observable characteristics (linked to r and/or Pareto weights). We can again identify an optimal mechanism (even with a partial revenue objective). Dworczak � Kominers R � Akbarpour R April 20, 2020 38
Redistribution through Markets Coming Soon: New Paper! We introduce quality q ; rate of substitution r now characterizes trade-off between q and money. Agents also have a label representing observable characteristics (linked to r and/or Pareto weights). We can again identify an optimal mechanism (even with a partial revenue objective). Assortative matching on r × q under revenue maximization, or when lump-sum transfer especially valuable. Random matching when poorer agents also value quality. Dworczak � Kominers R � Akbarpour R April 20, 2020 38
Redistribution through Markets Coming Soon: New Paper! We introduce quality q ; rate of substitution r now characterizes trade-off between q and money. Agents also have a label representing observable characteristics (linked to r and/or Pareto weights). We can again identify an optimal mechanism (even with a partial revenue objective). Assortative matching on r × q under revenue maximization, or when lump-sum transfer especially valuable. Random matching when poorer agents also value quality. More generally, regions of assortative and random matching. Dworczak � Kominers R � Akbarpour R April 20, 2020 38
Redistribution through Markets Coming Soon: New Paper! We introduce quality q ; rate of substitution r now characterizes trade-off between q and money. Agents also have a label representing observable characteristics (linked to r and/or Pareto weights). We can again identify an optimal mechanism (even with a partial revenue objective). Assortative matching on r × q under revenue maximization, or when lump-sum transfer especially valuable. Random matching when poorer agents also value quality. More generally, regions of assortative and random matching. Surprising(?) insight: Degree of assortativeness may be non-monotone in welfare weight on “poorer” agents. Dworczak � Kominers R � Akbarpour R April 20, 2020 38
Redistribution through Markets Coming Soon: New Paper! We introduce quality q ; rate of substitution r now characterizes trade-off between q and money. Agents also have a label representing observable characteristics (linked to r and/or Pareto weights). We can again identify an optimal mechanism (even with a partial revenue objective). Assortative matching on r × q under revenue maximization, or when lump-sum transfer especially valuable. Random matching when poorer agents also value quality. More generally, regions of assortative and random matching. Surprising(?) insight: Degree of assortativeness may be non-monotone in welfare weight on “poorer” agents. Why? Past some point, revenue is a better way to redistribute! Dworczak � Kominers R � Akbarpour R April 20, 2020 38
Redistribution through Markets Policy Dworczak � Kominers R � Akbarpour R April 20, 2020 39
Redistribution through Markets Policy Dworczak � Kominers R � Akbarpour R April 20, 2020 39
Redistribution through Markets Wrap Micro “market design” approach to redistribution. Dworczak � Kominers R � Akbarpour R April 20, 2020 40
Redistribution through Markets Wrap Micro “market design” approach to redistribution. If you regulate/control an individual market, it may be worth distorting allocative efficiency to improve equity. Dworczak � Kominers R � Akbarpour R April 20, 2020 40
Redistribution through Markets Wrap Micro “market design” approach to redistribution. If you regulate/control an individual market, it may be worth distorting allocative efficiency to improve equity. Rent control, price controls in kidney markets, and similar might be optimal responses to underlying inequality. (Complements macro approaches from PF!) Dworczak � Kominers R � Akbarpour R April 20, 2020 40
Redistribution through Markets Wrap Micro “market design” approach to redistribution. If you regulate/control an individual market, it may be worth distorting allocative efficiency to improve equity. Rent control, price controls in kidney markets, and similar might be optimal responses to underlying inequality. (Complements macro approaches from PF!) cross-side ineq. ⇒ price wedge + lump-sum redistribution same-side ineq. ⇒ rationing Dworczak � Kominers R � Akbarpour R April 20, 2020 40
Redistribution through Markets Wrap Micro “market design” approach to redistribution. If you regulate/control an individual market, it may be worth distorting allocative efficiency to improve equity. Rent control, price controls in kidney markets, and similar might be optimal responses to underlying inequality. (Complements macro approaches from PF!) cross-side ineq. ⇒ price wedge + lump-sum redistribution same-side ineq. ⇒ rationing ❀ Real opportunity for this type of design with the rise of new marketplace businesses. Dworczak � Kominers R � Akbarpour R April 20, 2020 40
Redistribution through Markets Wrap Micro “market design” approach to redistribution. If you regulate/control an individual market, it may be worth distorting allocative efficiency to improve equity. Rent control, price controls in kidney markets, and similar might be optimal responses to underlying inequality. (Complements macro approaches from PF!) cross-side ineq. ⇒ price wedge + lump-sum redistribution same-side ineq. ⇒ rationing ❀ Real opportunity for this type of design with the rise of new marketplace businesses. Framework for “Inequality-Aware” Marketplace Design ( v K , v M ) ❀ “Macro-Founded” Micro Dworczak � Kominers R � Akbarpour R April 20, 2020 40
Redistribution through Markets Wrap Micro “market design” approach to redistribution. If you regulate/control an individual market, it may be worth distorting allocative efficiency to improve equity. Rent control, price controls in kidney markets, and similar might be optimal responses to underlying inequality. (Complements macro approaches from PF!) cross-side ineq. ⇒ price wedge + lump-sum redistribution same-side ineq. ⇒ rationing ❀ Real opportunity for this type of design with the rise of new marketplace businesses. Framework for “Inequality-Aware” Marketplace Design ( v K , v M ) ❀ “Macro-Founded” Micro \ end { talk } Dworczak � Kominers R � Akbarpour R April 20, 2020 40
Redistribution Extra Slides Main Argument (I) ❀ Maximize over H S , H B ∈ ∆([0 , 1]) , U B , U S ≥ 0 � 1 � 1 µ φ α B ( q ) dH B ( q ) + φ α S ( q ) dH S ( q ) 0 0 subject to � 1 � 1 µ qdH B ( q ) = qdH S ( q ) . 0 0 Dworczak � Kominers R � Akbarpour R April 20, 2020 41
Redistribution Extra Slides Main Argument (II) ❀ Maximize over H S , H B ∈ ∆([0 , 1]) , U B , U S ≥ 0 � 1 µ φ α B ( q ) dH B ( q ) 0 � 1 �� G − 1 � ( q ) S (Π Λ + S ( r ) − α J S ( r )) g S ( r ) dr + (Λ S − α ) U S dH S ( q ) 0 0 Dworczak � Kominers R � Akbarpour R April 20, 2020 42
Redistribution Extra Slides Main Argument (II) ❀ Maximize over H S , H B ∈ ∆([0 , 1]) , U B , U S ≥ 0 � 1 µ φ α B ( q ) dH B ( q ) 0 � 1 �� G − 1 � ( q ) S (Π Λ + S ( r ) − α J S ( r )) g S ( r ) dr + (Λ S − α ) U S dH S ( q ) , 0 0 � where Λ j = λ j ( r ) dG j ( r ) is the average weight of j and � 1 � r r λ B ( r ) dG B ( r ) 0 λ S ( r ) dG S ( r ) Π Λ Π Λ B ( r ) := , S ( r ) := . g B ( r ) g S ( r ) Dworczak � Kominers R � Akbarpour R April 20, 2020 42
Redistribution Extra Slides Main Argument (II) ❀ Maximize over H S , H B ∈ ∆([0 , 1]) , U B , U S ≥ 0 � 1 µ φ α B ( q ) dH B ( q ) 0 � 1 �� G − 1 � ( q ) S (Π Λ + S ( r ) − α J S ( r )) g S ( r ) dr + (Λ S − α ) U S dH S ( q ) , 0 0 � where Λ j = λ j ( r ) dG j ( r ) is the average weight of j and � 1 � r r 1 dG B ( r ) 0 1 dG S ( r ) Π Λ Π Λ B ( r ) := , S ( r ) := . g B ( r ) g S ( r ) Dworczak � Kominers R � Akbarpour R April 20, 2020 42
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