Autobidder's Dilemma: Why More Sophisticated Autobidders Lead to Worse Auction Efficiency

Thank you for your insightful and detailed comments.

Slice-based model: first we'd like to note that our model is consistent with prior work on non-uniform bid scaling, e.g., "Non-uniform Bid-scaling and Equilibria for Different Auctions: An Empirical Study" by researchers at Google. In light of this, we believe this model already captures the essence of non-uniform bidding in reality. For example, our messages might inform a platform's decision of merging two channels or (not) splitting a channel into several, whenever practically feasible. On the other hand, another important consideration here is the tradeoff between the practicality of the model and the cleanliness of the messages. In principle, one could certainly consider more general / sophisticated models and try to derive similar results there (e.g., we sketch one such possibility in our response to Reviewer TSHw), but doing so would probably make the messages much more obscure. To this end, we'd like to argue that our approach achieves a reasonable balance on the spectrum.

Conclusions / messages of this paper: We position our paper as a paper providing a theoretical explanation of the empirical findings from "Non-uniform Bid-scaling and Equilibria for Different Auctions: An Empirical Study". The conclusion might seem unsurprising provided the previously known empirical results but we’d like to argue that it is indeed technically non-trivial to develop such a theoretical explanation. Our result is also more interpretable and reliable (in the sense that there is a proof) than empirical ones regarding the same phenomenon.

Thank you for your comments.

In words: all bidders are symmetric and use the same slices, which are given exogenously. Other than slices, we use the standard multi-item model, where essentially all items arrive at once at the very beginning, with the values given exogenously and publicly known (whether the values are public does not affect the behavior of the bidders, since they are responding to the bids of other bidders). Each bidder picks a bid multiplier for each slice, so each bidder's strategy is a vector of bid multipliers. We then consider equilibrium strategy profiles in this perfect-information environment, where each bidder's strategy must maximize their own total value subject to the ROI constraint, given all other bidders' strategies. Our measure of efficiency is the price of anarchy, i.e., the welfare under the worst equilibrium, divided by the first-best optimal welfare. We hope the above informal explanation can help clarify the reviewer's questions.

Thank you for your insightful and detailed comments.

Bidders have different slice partitions: first we'd like to note that our model is consistent with prior work on non-uniform bid scaling, and in particular, our results can be viewed as a theoretical explanation for the recent empirical study "Non-uniform Bid-scaling and Equilibria for Different Auctions: An Empirical Study" from Google Research -- they also consider environments where all bidders share the same partitions. In light of this, we believe this model already captures the essence of non-uniform bidding in reality. On the other hand, we do agree that a more general model with different partitions would be even better. In such a model, one can probably extend our results in the following way:

• Refine the definition of (un)balancedness. In particular, the reasonable definition of balancedness of a slice should exclude bidders who don't participate in auctions of this slice. In addition, the definition of unbalancedness should take into consideration the "adjacency" between bidders, where two bidders are adjacent if they are both active in some slice. The resulting definition would probably be based on a "balancedness graph", which degenerates to our definition when the graph is complete.
• Then, when establishing the bounds, one would follow the same high-level plan of forcing a worst-case equilibrium, but now under more constraints. More specifically, one would probably need to match slices that "generate" surplus to those that "consume" surplus, subject to the constraint that the surplus of each bidder flowing out of a slice cannot exceed that bidder's market share. One would need to find the worst-case matching that forces the worst welfare.

One important consideration here is the tradeoff between the complexity of the model and the cleanliness / readability of the results. The above plan for extending our results would probably result in a far more complicated bound that depends on more parameters of the market. Given this, we'd like to argue that our approach achieves a reasonable balance on the spectrum.

Intuitions on unbalancedness: the definition "naturally" arises in the process of trying to force a worst-case equilibrium. The idea is that low welfare in auctions with autobidders is typically because in some auctions, competition is extremely low, so the winner gets surplus almost for free. These bidders who have high surplus then spend the surplus in other auctions to compete with the "rightful winner", which ultimately hurts the welfare. One observation here is that when the latter phenomenon happens, the surplus "burnt" to beat the rightful winner consists of two parts: the part that one overpays for auctions where they are the rightful winner, and the part one overpays to beat the rightful winner. While the latter amount is always the same as the loss of welfare, the former does not contribute to this loss. So, to cause loss more efficiently, one would like to minimize the former amount. Fixing the total surplus burnt, this is done by burning more surplus on slices that are less balanced -- hence the definition of balancedness.

Now, another observation is that the two phenomena can hardly happen simultaneously in one slice, because the first requires a low bid multiplier, and the second requires a high one. So, to force the worst-case welfare, intuitively one would greedily allocate the most unbalanced slices to the second phenomenon and ensure that the rest of the slices generate just enough surplus for burning. This corresponds to an intricate optimization problem over a set of functions, which turns out to be controlled by the balancedness of each slice roughly in the way that the unbalancedness is defined.

Thank you for your insightful and encouraging comments.

Practicality of (per-slice) uniform bidding: while autobidders in reality are presumably more sophisticated, there is evidence that the advertising industry finds per-slice uniform bidding a reasonable approximation. In particular, the recent paper "Non-uniform Bid-scaling and Equilibria for Different Auctions: An Empirical Study" from Google Research considers a form of non-uniform bidding strategies essentially identical to per-slice uniform bidding. In their paper, they "define partitions to the queries" where "for each partition d, a non-uniform bid-scaling strategy chooses one bid multiplier", and study "how the simulation results vary with the granularity of the partition". Our results can be viewed as a theoretical explanation of their empirical findings. As such, we'd argue that conceptually, our results have similar practical implications.