Differentiable Economics - David C. Parkes Harvard University - CityU CS
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Differentiable Economics
David C. Parkes
Harvard University
Joint work with G. Brero, D. Chakrabarti, P. Dütting, A. Eden, Z. Feng, M. Gerstgrasser,
N. Golowich, S. Li, V. Li, S. Kominers, J. Ma, H. Narasimhan, S. S. Ravindranath,
D. Rheingans-Yoo, and R. Trivedi
July 13, 2021
Inverse Problems
I Revenue optimal auction design
I Matching market design
I Optimal taxation policy design
I Contract design
I ...
Differentiable Economics 2 / 36
Typical Recipe
Solve via revelation principle (for valuation profile v, density
function f , exp utility U ):
Z X
max ti (v)f (v)dv
g,t v i
s.t. feasible g
Ui (g, t, vi ) ≥ 0
Ui (g, t, vi ) ≥ Ui (g, t, v̂i ), ∀i, ∀vi , ∀v̂i
Allocation g : V → [0, 1]n , payment t : V → Rn
Differentiable Economics 3 / 36
An Optimal 1-Bidder, 2-item Auction
Two additive items i.i.d. U (0, 1). Manelli-Vincent (2006)
value item 2
1
(0, 1)
2
3 (1, 1)
y
√
2− 2 (0, 0)
3
(1, 0) value
x item 1
Differentiable Economics 4 / 36
An Optimal 1-Bidder, 2-item Auction
Two additive items i.i.d. U (0, 1). Manelli-Vincent (2006)
value item 2
1
(0, 1)
2
3 (1, 1)
y
√
2− 2 (0, 0)
3
(1, 0) value
x item 1
Differentiable Economics 4 / 36
A Role for Computation
I Automated mechanism design (CS02)
I Standard approaches:
I Dominant strategy incentive compatible, but via LPs and fail
to scale or handle continue types
I Bayesian incentive compatible (HL10, CDW12, AFH+12,...)
Differentiable Economics 5 / 36
A Role for Computation
I Automated mechanism design (CS02)
I Standard approaches:
I Dominant strategy incentive compatible, but via LPs and fail
to scale or handle continue types
I Bayesian incentive compatible (HL10, CDW12, AFH+12,...)
Differentiable Economics 5 / 36
This work: Use Neural Networks
Model the rules of an economic system as a flexible,
differentiable representation. Differentiable economics.
Differentiable Economics 6 / 36
Talk Outline: Three Vignettes
I Revenue-optimal auction design
I Two-sided matching market design
I Indirect mechanism design (sequential price mechanisms)
Differentiable Economics 7 / 36
Part I: Revenue-Optimal Auction Design
“Optimal Auctions through Deep Learning,” Dütting, Feng,
Narasimhan, Parkes, Ravindranath, Proc. ICML’19
I A seller with m distinct indivisible items
I A set of n(≥ 1) additive buyers, with valuations
vi = (vi1 , . . . , vim ), and vi ∼ Fi
I Design an auction (g w , tw ) that maximizes expected revenue
s.t. dominant-strategy IC
I parameters w, allocation rule g w , payment rule tw
Differentiable Economics 8 / 36Part I: Revenue-Optimal Auction Design
“Optimal Auctions through Deep Learning,” Dütting, Feng,
Narasimhan, Parkes, Ravindranath, Proc. ICML’19
I A seller with m distinct indivisible items
I A set of n(≥ 1) additive buyers, with valuations
vi = (vi1 , . . . , vim ), and vi ∼ Fi
I Design an auction (g w , tw ) that maximizes expected revenue
s.t. dominant-strategy IC
I parameters w, allocation rule g w , payment rule tw
Differentiable Economics 8 / 36Architecture 1: RochetNet (Single Bidder)
I Learn a menu set {hj }. Utility for choice j, hj (b) = αj · b − βj
I Parameters for jth choice define a randomized allocation
αj ∈ [0, 1]m and a payment βj ∈ R
I Train to maximize expected revenue
Differentiable Economics 9 / 36Architecture 1: RochetNet (Single Bidder)
I Learn a menu set {hj }. Utility for choice j, hj (b) = αj · b − βj
I Parameters for jth choice define a randomized allocation
αj ∈ [0, 1]m and a payment βj ∈ R
I Train to maximize expected revenue
Differentiable Economics 9 / 36Architecture 1: RochetNet (Single Bidder)
I Learn a menu set {hj }. Utility for choice j, hj (b) = αj · b − βj
I Parameters for jth choice define a randomized allocation
αj ∈ [0, 1]m and a payment βj ∈ R
I Train to maximize expected revenue
Differentiable Economics 9 / 36Manelli-Vincent setting
value item 2
1
I Value 1 v1 ∼ U (0, 1) 2
(0, 1)
(1, 1)
3
I Value 2 v2 ∼ U (0, 1) √
y
2− 2
3
(0, 0)
(1, 0) value
x item 1
Differentiable Economics 10 / 36Manelli-Vincent setting
value item 2
1
I Value 1 v1 ∼ U (0, 1) 2
(0, 1)
(1, 1)
3
I Value 2 v2 ∼ U (0, 1) √
y
2− 2
3
(0, 0)
(1, 0) value
x item 1
Prob. of allocating item 1 Prob. of allocating item 2
1.0 1.0 1.0 1.0
0.8 0.8 0.8 0.8
1 1
0.6 0.6 0.6 0.6
v2
v2
0.4 0.4 0.4 0.4
0 0
0.2 0.2 0.2 0.2
0.0 0.0 0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
v1 v1
Differentiable Economics 10 / 36Architecture 2: RegretNet (Multi Bidder)
m items, n additive bidders, bid bij of agent i for item j
Allocation Payment
tw : Rnm → Rn≥0
g w : Rnm → ∆1 × · · · × ∆m αi is fraction of value charged to i
Differentiable Economics 11 / 36RegretNet Training Problem (1 of 2)
" n
#
X
min −Ev tw
i (v)
w
i=1
s.t. Ev [regret w
i (v)] = 0, all bidders i
regret w
i (v) is maximum utility gain to bidder i from misrport at
profile v. No training labels!
Solve via (augmented) Lagrangian method:
" n # n
X X
w
min −Ev ti (v) + λi · Ev [regret w
i (v)] + · · ·
w
i=1 i=1
Differentiable Economics 12 / 36RegretNet Training Problem (1 of 2)
" n
#
X
min −Ev tw
i (v)
w
i=1
s.t. Ev [regret w
i (v)] = 0, all bidders i
regret w
i (v) is maximum utility gain to bidder i from misrport at
profile v. No training labels!
Solve via (augmented) Lagrangian method:
" n # n
X X
w
min −Ev ti (v) + λi · Ev [regret w
i (v)] + · · ·
w
i=1 i=1
Differentiable Economics 12 / 36RegretNet Training Problem (1 of 2)
" n
#
X
min −Ev tw
i (v)
w
i=1
s.t. Ev [regret w
i (v)] = 0, all bidders i
regret w
i (v) is maximum utility gain to bidder i from misrport at
profile v. No training labels!
Solve via (augmented) Lagrangian method:
" n # n
X X
w
min −Ev ti (v) + λi · Ev [regret w
i (v)] + · · ·
w
i=1 i=1
Differentiable Economics 12 / 36RegretNet Training Problem (2 of 2)
Adopt stochastic gradient descent to solve the (augmented)
Lagrangian optimization problem:
" n # n
X X
w
min −Ev ti (v) + λi · Ev [regret w
i (v)] + · · ·
w
i=1 i=1
Key challenge: taking derivative through the inner maximization
0
regret w
i (v) = max
0
[uw w
i (vi , v−i ) − ui (vi , v−i )]
vi
Idea: fix a ‘defeating misreport’ for (i, v), found via gradient
ascent on input
Differentiable Economics 13 / 36RegretNet Training Problem (2 of 2)
Adopt stochastic gradient descent to solve the (augmented)
Lagrangian optimization problem:
" n # n
X X
w
min −Ev ti (v) + λi · Ev [regret w
i (v)] + · · ·
w
i=1 i=1
Key challenge: taking derivative through the inner maximization
0
regret w
i (v) = max
0
[uw w
i (vi , v−i ) − ui (vi , v−i )]
vi
Idea: fix a ‘defeating misreport’ for (i, v), found via gradient
ascent on input
Differentiable Economics 13 / 36RegretNet Training Problem (2 of 2)
Adopt stochastic gradient descent to solve the (augmented)
Lagrangian optimization problem:
" n # n
X X
w
min −Ev ti (v) + λi · Ev [regret w
i (v)] + · · ·
w
i=1 i=1
Key challenge: taking derivative through the inner maximization
0
regret w
i (v) = max
0
[uw w
i (vi , v−i ) − ui (vi , v−i )]
vi
Idea: fix a ‘defeating misreport’ for (i, v), found via gradient
ascent on input
Differentiable Economics 13 / 36Gradient ascent to find defeating misreports
I Green dot = true valuation, Red dots = misreports
Differentiable Economics 14 / 36Manelli-Vincent setting
value item 2
1
I Value 1 v1 ∼ U (0, 1) 2
(0, 1)
(1, 1)
3
I Value 2 v2 ∼ U (0, 1) √
y
2− 2
3
(0, 0)
(1, 0) value
x item 1
Differentiable Economics 15 / 36RegretNet: 2 bidders, 2 items, discrete
values
Extends Yao (2017). For each bidder:
I Item 1 vi,1 ∼unif {0.5, 1, 1.5}
I Item 2 vi,2 ∼unif {0.5, 1, 1.5}
Differentiable Economics 16 / 36RegretNet: {3, 5}×10 problems
I Additive U (0, 1) values
I Item-wise Myerson is optimal in limit of large n (Palfrey’83)
I Use a larger 5x100 neural network
Differentiable Economics 17 / 36Theory
I Generalization bounds for expected revenue and regret
I Discover provably optimal designs via RochetNet
I Support conjectures: optimality for m ≥ 7 of “straight-jacket
auction” for additive U (0, 1) items (GK’18):
Items SJA (rev ) RochetNet (rev )
2 0.549187 0.549175
6 1.943239 1.943216
7 2.318032 2.318032
9 3.086125 3.086125
10 3.477781 3.477722
Differentiable Economics 18 / 36Theory
I Generalization bounds for expected revenue and regret
I Discover provably optimal designs via RochetNet
I Support conjectures: optimality for m ≥ 7 of “straight-jacket
auction” for additive U (0, 1) items (GK’18):
Items SJA (rev ) RochetNet (rev )
2 0.549187 0.549175
6 1.943239 1.943216
7 2.318032 2.318032
9 3.086125 3.086125
10 3.477781 3.477722
Differentiable Economics 18 / 36Theory
I Generalization bounds for expected revenue and regret
I Discover provably optimal designs via RochetNet
I Support conjectures: optimality for m ≥ 7 of “straight-jacket
auction” for additive U (0, 1) items (GK’18):
Items SJA (rev ) RochetNet (rev )
2 0.549187 0.549175
6 1.943239 1.943216
7 2.318032 2.318032
9 3.086125 3.086125
10 3.477781 3.477722
Differentiable Economics 18 / 36Talk Outline: Three Vignettes
I Revenue-optimal auction design
I Two-sided matching market design
I Indirect mechanism design (sequential price mechanisms)
Differentiable Economics 19 / 36Part II: Two-sided matching markets
Workers W , Firms F . Strict preferences. For example:
f2 w1 f1 w1 f3 w1 f1 w3 f1 w2
f1 w2 f3 w2 f2 w3 f2 w1 f2 w2
f1 w3 f2 w3 f3 w1 f3 w3 f3 w2
Matching (a) is unstable; e.g., (w1 , f2 ) is a blocking pair.
Matching (b) is stable.
Differentiable Economics 20 / 36Trading-off Stability and Strategy-proofness
I S. S. Ravindranath, Z. Feng, S. Li, J. Ma, S. Kominers and
D. C. Parkes, arXiv 2021
I Impossibility theorem (Roth’82): There is no mechanism
that is stable and strategy-proof
I But little is understood about how to make tradeoffs. We
have only a point solution (deferred acceptance).
Differentiable Economics 21 / 36The Matching Network
column − wise
normalization
β
× s̄11 ... s̄1m
p11 s11 σ+ r11 = min{ŝ11 , ŝ011 }
(1) (R)
.. h1 h1 .. .. ..
. . . β s̄n1 ... s̄nm .
ŝ
pnm sn+1m σ+ rn1 = min{ŝn1 , ŝ0n1 }
(1) (R) ×
h2 h2 s̄⊥1 .. s̄⊥m
... . ..
ŝ0 .
.. .. β
q11 . . s011 σ+ r1m = min{ŝ1m , ŝ01m }
× s̄011 ... s̄01m s̄01⊥
.. .. .. ..
. (1)
hJ1
(R)
hJR . . β ... .
qnm s0nm+1 σ+
... rnm = min{ŝnm , ŝ0nm }
× s̄0n1 s̄0nm s̄0n⊥
row − wise
normalization
4 × 256 network + additional layers. Cardinal inputs; e.g.,
pw· = ( 31 , 23 , − 31 ) for f2 w f1 w ⊥ w f3 .
Output is the marginal probabilities for pairwise matches (0-1
decomposition via Birkhoff von-Neumann)
Differentiable Economics 22 / 36The Training Problem
Loss function:
min λ · stv (g w ) + (1 − λ) · rgt(g w ), for λ ∈ [0, 1]
w
I stv (g w ) quantifies the violation of a generalization of stability
that is suitable for randomized matchings
I rgt(g w ) quantifies the expected regret, similarly to
RegretNet (can extend to ordinal SP)
Differentiable Economics 23 / 36The Design Frontier
Loss function: minw λ · stv (g θ ) + (1 − λ) · rgt(g θ )
Correlated preferences; 4x4 market. New targets for theory!
pcorr=0.25
0.005 DA
= 16
64
0.004
SP-violation
8
= 64
0.003 6
= 64
4
0.002 = 64
3
= 64
0.001 2
= 64 RSD
0.000
0.00 0.05 0.10 0.15 0.20
Stability Violation
Differentiable Economics 24 / 36Talk Outline: Three Vignettes
I Revenue-optimal auction design
I Two-sided matching market design
I Indirect mechanism design (sequential price mechanisms)
Differentiable Economics 25 / 36Part III: Indirect Mechanism Design
I Many inverse problems are indirect
I Economic rules that induce a game that is played by
participants
I c.f., work on learning optimal taxation policies (Zheng et al.,
arXiv 2020)
Joint work with G. Brero, D. Chakrabarti, A. Eden, M. Gerstgrasser,
V. Li, and D. Rheingans-Yoo (AAAI’21, ICML’21 workshop, arXiv)
Differentiable Economics 26 / 36Illustration: Sequential Price Mechanisms
Differentiable Economics 27 / 36Illustration: Sequential Price Mechanisms
$5 $2
?
Differentiable Economics 27 / 36Illustration: Sequential Price Mechanisms
$4 $1.5
?
Differentiable Economics 27 / 36Illustration: Sequential Price Mechanisms
$1
?
Differentiable Economics 27 / 36Illustration: Sequential Price Mechanisms
Differentiable Economics 27 / 36Introducing Message passing
Differentiable Economics 28 / 36Introducing Message passing
Differentiable Economics 28 / 36Introducing Message passing
Differentiable Economics 28 / 36New Challenge: Agent Behavior
I Two-level learning: agents learn to play the game,
mechanism learns how to design the rules given this
I Formulate the AMD problem as a MultiFollower Stackelberg
game:
I Mechanism is a policy network from observations
(messages, purchases) to actions (next agent, prices);
strategy of leader = design of mechanism
I Agents are followers, learn to play the induced game
I In the special case of no communication, agents have a
dominant strategy equilibrium in SPMs
Differentiable Economics 29 / 36New Challenge: Agent Behavior
I Two-level learning: agents learn to play the game,
mechanism learns how to design the rules given this
I Formulate the AMD problem as a MultiFollower Stackelberg
game:
I Mechanism is a policy network from observations
(messages, purchases) to actions (next agent, prices);
strategy of leader = design of mechanism
I Agents are followers, learn to play the induced game
I In the special case of no communication, agents have a
dominant strategy equilibrium in SPMs
Differentiable Economics 29 / 36New Challenge: Agent Behavior
I Two-level learning: agents learn to play the game,
mechanism learns how to design the rules given this
I Formulate the AMD problem as a MultiFollower Stackelberg
game:
I Mechanism is a policy network from observations
(messages, purchases) to actions (next agent, prices);
strategy of leader = design of mechanism
I Agents are followers, learn to play the induced game
I In the special case of no communication, agents have a
dominant strategy equilibrium in SPMs
Differentiable Economics 29 / 36Max-Min Fairness (no communicaton)
Setting: 9 agents, 2 different kinds of items (5 units each). Train
via the PPO actor-critic, policy-gradient algorithm (S+17)
Differentiable Economics 30 / 36Two-Stage Learning
I Adopt multiplicate weights (MW) for agents to learn a
Bayesian coarse-correlated equilibrium
I Formulate a Stackelberg MDP. A single-player POMDP with
two phases:
I A learning phase, with Markovian learning-dynamics (MW)
representing adaptation of agents
I A best-response phase, with the reward to the leader based
on learned agent strategies
Also with R. Trivedi
Differentiable Economics 31 / 36Two-Stage Learning
I Adopt multiplicate weights (MW) for agents to learn a
Bayesian coarse-correlated equilibrium
I Formulate a Stackelberg MDP. A single-player POMDP with
two phases:
I A learning phase, with Markovian learning-dynamics (MW)
representing adaptation of agents
I A best-response phase, with the reward to the leader based
on learned agent strategies
Also with R. Trivedi
Differentiable Economics 31 / 36Two-Stage Learning
I Adopt multiplicate weights (MW) for agents to learn a
Bayesian coarse-correlated equilibrium
I Formulate a Stackelberg MDP. A single-player POMDP with
two phases:
I A learning phase, with Markovian learning-dynamics (MW)
representing adaptation of agents
I A best-response phase, with the reward to the leader based
on learned agent strategies
Also with R. Trivedi
Differentiable Economics 31 / 36Illustrative results
I 2 agents, 1 item. Agent 1’s value is {0, 1} w.p. {1/2, 1/2}, agent
2’s value is {1/2, 1/2} w.p. {1 − , }.
I Optimal SPM visits 1 then 2 (price zero), and inefficient when
v1 = 1, v2 = 1/2.
Differentiable Economics 32 / 36Discussion
I Direct mechanisms
I Scaling up; e.g., one agent for multiple market sizes
(RJBW’21), combinatorial problems
I Leveraging chatacterization results such as monotonicity
I Robustness, for example certificates of low regret
(CCGD’20)
I Indirect mechanisms
I Scaling up the Stackelberg MDP
I Leveraging gradient dynamics convergence for potential
games (BFHKS’21, MRS’20)
I Derivatives through equilibrium (WXPRT’21)
I Centralized learning, decentralized execution inspired
approaches (FAFW’16)
Differentiable Economics 33 / 36Discussion
I Direct mechanisms
I Scaling up; e.g., one agent for multiple market sizes
(RJBW’21), combinatorial problems
I Leveraging chatacterization results such as monotonicity
I Robustness, for example certificates of low regret
(CCGD’20)
I Indirect mechanisms
I Scaling up the Stackelberg MDP
I Leveraging gradient dynamics convergence for potential
games (BFHKS’21, MRS’20)
I Derivatives through equilibrium (WXPRT’21)
I Centralized learning, decentralized execution inspired
approaches (FAFW’16)
Differentiable Economics 33 / 36Conclusion
I End-to-end learning for economic design with
in-expectation objectives
I Flexible: seems interesting for a myriad of design problems
I Differentiable Economics
Thank You!
I Funding support: DARPA cooperative agreement HR00111920029; AWS gift
I P. Dütting, Z. Feng, H. Narasimhan, D. C. Parkes, S. S. Ravindranath: “Optimal Auctions through Deep Learning,”
ICML’19 (first version 2017); longer version arXiv 2020
I N. Golowich, H. Narasimhan, and D. C. Parkes: “Deep Learning for Multi-Facility Location Mechanism Design,” IJCAI’18
I Z. Feng, H. Narasimhan, and D. C. Parkes: “Deep Learning for Revenue-Optimal Auctions with Budgets,” AAMAS’18
I G. Brero, A. Eden, M. Gerstgrasser, D. C. Parkes, D. Rheingans-Yoo: “Reinforcement learning of simple indirect
mechanisms”, AAAI’21
I S. Zheng, A. Trott, S. Srinivasa, N. Naik, M. Gruesbeck, D. C. Parkes, R. Socher: “The AI Economist: Improving Equality
and Productivity with AI-Driven Tax Policies”, arXiv 2020
I Z. Feng, D. C. Parkes, S. S. Ravindranath: “Machine Learning for Matching Markets”, Online and Matching-Based
Market Design, Echenique et al. (Eds) 2021
I G. Brero, D. Chakrabarti, A. Eden, M. Gerstgrasser, V. Li, D. C. Parkes, “Learning Stackelberg Equilibria in Sequential
Price Mechanisms”, ICML 2021 Workshop on Reinforcement Learning Theory
Differentiable Economics 34 / 36Conclusion
I End-to-end learning for economic design with
in-expectation objectives
I Flexible: seems interesting for a myriad of design problems
I Differentiable Economics
Thank You!
I Funding support: DARPA cooperative agreement HR00111920029; AWS gift
I P. Dütting, Z. Feng, H. Narasimhan, D. C. Parkes, S. S. Ravindranath: “Optimal Auctions through Deep Learning,”
ICML’19 (first version 2017); longer version arXiv 2020
I N. Golowich, H. Narasimhan, and D. C. Parkes: “Deep Learning for Multi-Facility Location Mechanism Design,” IJCAI’18
I Z. Feng, H. Narasimhan, and D. C. Parkes: “Deep Learning for Revenue-Optimal Auctions with Budgets,” AAMAS’18
I G. Brero, A. Eden, M. Gerstgrasser, D. C. Parkes, D. Rheingans-Yoo: “Reinforcement learning of simple indirect
mechanisms”, AAAI’21
I S. Zheng, A. Trott, S. Srinivasa, N. Naik, M. Gruesbeck, D. C. Parkes, R. Socher: “The AI Economist: Improving Equality
and Productivity with AI-Driven Tax Policies”, arXiv 2020
I Z. Feng, D. C. Parkes, S. S. Ravindranath: “Machine Learning for Matching Markets”, Online and Matching-Based
Market Design, Echenique et al. (Eds) 2021
I G. Brero, D. Chakrabarti, A. Eden, M. Gerstgrasser, V. Li, D. C. Parkes, “Learning Stackelberg Equilibria in Sequential
Price Mechanisms”, ICML 2021 Workshop on Reinforcement Learning Theory
Differentiable Economics 34 / 36Conclusion
I End-to-end learning for economic design with
in-expectation objectives
I Flexible: seems interesting for a myriad of design problems
I Differentiable Economics
Thank You!
I Funding support: DARPA cooperative agreement HR00111920029; AWS gift
I P. Dütting, Z. Feng, H. Narasimhan, D. C. Parkes, S. S. Ravindranath: “Optimal Auctions through Deep Learning,”
ICML’19 (first version 2017); longer version arXiv 2020
I N. Golowich, H. Narasimhan, and D. C. Parkes: “Deep Learning for Multi-Facility Location Mechanism Design,” IJCAI’18
I Z. Feng, H. Narasimhan, and D. C. Parkes: “Deep Learning for Revenue-Optimal Auctions with Budgets,” AAMAS’18
I G. Brero, A. Eden, M. Gerstgrasser, D. C. Parkes, D. Rheingans-Yoo: “Reinforcement learning of simple indirect
mechanisms”, AAAI’21
I S. Zheng, A. Trott, S. Srinivasa, N. Naik, M. Gruesbeck, D. C. Parkes, R. Socher: “The AI Economist: Improving Equality
and Productivity with AI-Driven Tax Policies”, arXiv 2020
I Z. Feng, D. C. Parkes, S. S. Ravindranath: “Machine Learning for Matching Markets”, Online and Matching-Based
Market Design, Echenique et al. (Eds) 2021
I G. Brero, D. Chakrabarti, A. Eden, M. Gerstgrasser, V. Li, D. C. Parkes, “Learning Stackelberg Equilibria in Sequential
Price Mechanisms”, ICML 2021 Workshop on Reinforcement Learning Theory
Differentiable Economics 34 / 36Additional References (1 of 2)
I J. Rahme, S. Jelassi, J. Bruna, M. Weinberg, “A Permutation-Equivariant Neural Network Architecture For Auction
Design” AAAI’21
I M. J. Curry, P. Chiang, T. Goldstein, J. Dickerson, “Certifying Strategyproof Auction Networks” NeurIPS’20
I S. Zheng, A. Trott, S. Srinivasa, N. Naik, M. Gruesbeck, D. C. Parkes, R. Socher, “The AI Economist: Improving Equality
and Productivity with AI-Driven Tax Policies” arXiv 2004.13332, 2020
I M. Bichler, M. Fichtl, S. Heidekrüger, N. Kohring, P. Sutterer, “Learning Equilibria in Symmetric Auction Games using
Artificial Neural Networks,” Nature Machine Intelligence (forthcoming)
I E. Mazumdar, L. J. Ratliff, S. S. Sastry, “On gradient-based learning in continuous games,” SIAM Journal on
Mathematics of Data Science 2020.
I K. Wang, L. Xu, A. Perrault, M. K. Reiter, M. Tambe, “Coordinating Followers to Reach Better Equilibria: End-to-End
Gradient Descent for Stackelberg Games” arXiv 2021
I J. Foerster, I. Assael, N. de Freitas, S. Whiteson, “Learning to communicate with deep multi-agent reinforcement
learning” NeurIPS 2016
I J. Hartline, V. Syrgkanis, E. Tardos, “No-regret learning in Bayesian games,” arXiv 2015
I J. Schulman, F. Wolski, P. Dhariwal, A. Radford, O. Klimov, “Proximal Policy Optimization Algorithms”, arXiv 2017
I J. D. Hartline, B. Lucier, “Bayesian algorithmic mechanism design” STOC 2010
I Y. Cai, C. Daskalakis, M. S. Weinberg, “Optimal multi-dimensional mechanism design: Reducing revenue to welfare
maximization” FOCS’12
I S. Alaei, H. Fu, N. Haghpanah, J. D. Hartline, A. Malekian, “Bayesian optimal auctions via multi- to single-agent
reduction” EC’12
I V. Conitzer, T. Sandholm, “Complexity of mechanism design” UAI’02
Differentiable Economics 35 / 36Additional References (2 of 2)
I A. Manelli, D. Vincent, “Bundling as an optimal selling mechanism for a multiple-good monopolist” Journal of Economic
Theory 2006.
I A. C.-C. Yao, “Dominant-strategy versus bayesian multi-item auctions: Maximum revenue determination and
comparison” EC’17
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