## Abstract

This position paper anticipates ways in which the disruptive developments in machine learning over the past few years could be leveraged for a new generation of computational methods that automate the process of designing optimal economic mechanisms.

We would like to thank Yang Cai, Vincent Conitzer, Constantinos Daskalakis, Scott Kominers, Alexander Rush, and participants at seminars at the Simons Institute for the Theory of Computing, Dagstuhl, London School of Economics, IJCAI’18, AAMAS’18, The Technion, the EC’18 WADE workshop, the German Economic Association annual meeting, Google, MIT/Harvard Theory Seminar, WWW’19 workshop, and HBS for their helpful feedback.

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## Notes

- 1.
Results are known for additive i.i.d.

*U*(0, 1) values on items (Manelli and Vincent 2006), additive, independent and asymmetric distributions on item values (Daskalakis et al. 2017; Giannakopoulos and Koutsoupias 2015; Thirumulanathan et al. 2016), additive, i.i.d. exponentially distributed item values (Daskalakis et al. 2017) and extended to multiple items (Giannakopoulos 2015), additive, i.i.d. Pareto distributions on item values (Hart and Nisan 2012), unit-demand valuations with item values i.i.d. \(U(c,c+1)\), \(c>0\) (Pavlov 2011), and unit-demand, independent, uniform and asymmetric distributions on item values (Thirumulanathan et al. 2017). - 2.
Working in increasingly general settings, relevant results on DSIC auction design include Chawla et al. (2007, 2010), Alaei (2014), Kleinberg and Weinberg (2012), Hart and Nisan (2012), Li and Yao (2013), Babaioff et al. (2014), Yao (2015), Rubinstein and Weinberg (2015), Cai et al. (2016), Cai and Zhao (2017), Dütting et al. (2017). These mechanisms are simple, and reveal the structural ingredients that are important for the design of mechanisms with good revenue properties.

- 3.
IC means that no agent can benefit, in equilibrium, by misreporting its type, and can hold in a dominant-strategy equilibrium (DSIC) or a Bayes-Nash equilibrium (BIC). We will generally be interested in DSIC.

- 4.
See Parkes and Wellman (2015) for a discussion on the role of AI in the mediation of economic transactions.

- 5.
- 6.
For example, we could also penalize failure of weak-monotonicity (Bikhchandani et al. 2006), or insist that the implied pricing-function is agent independent (with prices to an agent that are do not depend on its report, conditioned on an allocation).

- 7.
Daskalakis et al. (2014) give a complexity result for optimal mechanism design. There is also a recent literature on the sample complexity of auctions and mechanisms, including revenue-optimal auctions (Elkind 2007; Cole and Roughgarden 2014; Dughmi et al. 2014; Morgenstern and Roughgarden 2015, 2016; Huang et al. 2015; Devanur et al. 2016; Narasimhan and Parkes 2016; Gonczarowski and Nisan 2017; Cai and Daskalakis 2017).

- 8.
The use of machine learning for mechanism design was earlier pioneered by Dütting et al. (2015), who use support vector machines to design payment rules for a given allocation rule (which can be designed to be scalable). But their framework can fail to even closely approximate incentive compatibility then the rule is not implementable, and does not support design objectives that are stated on payments. Earlier, Procaccia et al. (2009) studied the learnability of specific classes of voting rules, but without considering incentives; see also Xia (2013), who suggests a learning framework that incorporates specific axiomatic properties.

- 9.
Deep learning, which refers typically to the use of multi-layer neural networks, has gained a great deal of attention in recent years. This is because of the existence of large data sets, the development of tool chains that make experimentation easy, optimized hardware to speed-up training (GPUs), as well as massive investment from the private sector. Whether a network is considered ‘deep’ or not is a matter of taste.

- 10.
- 11.
The sigmoid activation function is \(\sigma (z)\,{=}\,1/(1+e^{-z})\). The softmax activation function for item

*j*is \( softmax _i(s_{1j}, \ldots , s_{nj}, s_{n+1,j})\,{=}\, e^{s_{ij}}/\sum _{k=1}^{n+1} e^{s_{kj}}\), where \(s_{n+1,j}\) is a dummy input that corresponds to the item not being allocated to any bidder. In another variation, we handle unit-demand valuations of bidders by using an additional set of softmax activation functions, one per agent, and taking the minimum of these item-wise and agent-wise softmax components in defining the output layer. - 12.
The output \(p_i\) that corresponds to bidder

*i*is the amount the bidder should pay in expectation, for a particular bid profile. This can be converted into an equivalent lottery on payments, such that a bidder’s payment is no greater than her value for any realized allocation (the property of*ex post*IR). - 13.
In more recent work (Dütting et al. 2019) we take an adversarial-style approach, using a gradient-based approach for estimating regret for a given profile. The gradient-based approach requires that the valuation space is continuous and the utility function is differentiable, but is more scalable and stable for larger settings.

- 14.
With a suitably large penalty parameter \(\rho \), the method of augmented Lagrangian is guaranteed to converge to a (locally) optimal solution to the original problem (Wright and Nocedal 1999). In practice we find that even for small values of \(\rho \) and enough iterations, the solver converges to auction designs that yield near-optimal revenue while closely satisfying the regret constraints.

- 15.
A training iteration is one mini-batch gradient updated in the solver that we use for stochastic gradient descent.

- 16.
Pai and Vohra (2014) design the optimal, single-item BIC auction, while Malakhov and Vohra (2008) provide the state-of-the-art result for the optimal, single-item DSIC auction (for two bidders, and with a weaker “conditional” form of DSIC). These results build on earlier results for more stylized settings (Che and Gale 1998, 2000; Maskin 2000; Laffont and Robert 1996). There are also a few approximation results for DSIC and BIC designs (Borgs et al. 2005; Bhattacharya et al. 2012; Chawla et al. 2011).

- 17.
Ehlers and Gordon (2011) and Heo (2013) provide characterizations for the special case of \(K=2\) under additional assumptions. Heo (2013) assumes anonymity and an additional property,

*users only*, which means that agents cannot influence the locations of facilities they will not use. Ehlers and Gordon (2011) assume that agents have*lexmax*preferences over facilities, and thus do not only care about the peak closest to them. - 18.
For the single facility location problem, it is w.l.o.g. to consider mechanisms that operate on agent peaks (Border and Jordan 1983). This extends also to a more general “voting under constraints” setting (Barberà et al. 1997). For multiple facilities there are DSIC mechanisms that do not depend only on agent peaks. For example, one can consider the example of \(K=2\) facilities and \(n=2\), where one facility is placed at the peak of agent 1 and the other at some location that depends on the shape of agent 1’s report. Still, we retain this simple representation in our current work.

- 19.
In a multi-facility, dictatorial rule, each facility is determined by a separate dictatorial rule. This is equivalent to having a serial-dictatorial rule for all

*K*facilities. In a multi-facility, percentile rule, each facility is determined by a separate percentile rule (Sui et al. 2013). A constant rule places each facility in the same location all the time. - 20.
If \(p_1, \ldots , p_n\) are the agent peaks in sorted order, then a facility at location

*x*has percentile 0 if \(x \le p_1\), has percentile 1 if \(x \ge p_n\), and has percentile \(\frac{i-1}{n-1} + \frac{x - p_i}{(n-1)(p_{i+1} - p_i)}\) if \(p_i \le x < p_{i+1}\).

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## Acknowledgements

This research is supported in part by NSF EAGER #124110.

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Dütting, P., Feng, Z., Golowich, N., Narasimhan, H., Parkes, D.C., Ravindranath, S.S. (2019). Machine Learning for Optimal Economic Design. In: Laslier, JF., Moulin, H., Sanver, M., Zwicker, W. (eds) The Future of Economic Design. Studies in Economic Design. Springer, Cham. https://doi.org/10.1007/978-3-030-18050-8_70

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