Common use of Empirical Results Clause in Contracts

Empirical Results. We implemented our algorithm and experiments in Python 2.7, using the Glop linear programming solver1 to compute the liquid welfare within our algorithm. Each run of the ex- periment takes roughly 30 seconds on a single CPU. Figure 1 demonstrates the revenue and welfare trends for a single in- ventory unit as the budget ratio r is varied (trends were very similar across all inventory units), and Table 1 summarizes the results for r = 1. From Figure 1 we see that our algo- rithm’s (AAGBudget) revenue performance closely tracks the optimal revenue achievable as captured by the liquid welfare for all values of r. The original AAG algorithm that does not take budget into account (AAGNoBudget) performs poorly especially when the budgets are small, and improves as the budgets increase. This is because, when the budget is small, the AAGNoBudget algorithm might offer a deal that violates buyer’s budget. Given such a deal, the buyer may reject the deal, resulting in 0 revenue. This is why AAGBudget shows 0 revenue up to r = 0.5. Recall that for AAGBudget and AAGNoBudget, revenue is equivalent to welfare. An opposite revenue trend holds for the second-price auc- tion benchmarks: their revenue performance decreases as r increases. This is because when the budget is small, the second price auction may be able to exhaust all the budgets within 100K auctions and therefore approach the liquid wel- fare optimum. For second-price auctions, especially the one with optimal reserves, there is a trade-off between revenue and efficiency. To understand why welfare can be higher than liquid welfare for the second-price auctions, recall that liquid 1See ▇▇▇▇▇://▇▇▇▇▇▇▇▇▇▇.▇▇▇▇▇▇.▇▇▇/optimization/lp/glop. welfare only provides an upper bound on the revenue but not the welfare. If auction prices are consistently low but values are high, it is possible to achieve a high total welfare beyond the available budget, while respecting budget constraints. Table 1 provides results, indexed against the optimal so- cial welfare (unconstrained by budget) to interpret both the revenue and social welfare levels together. When compar- ing revenue directly against liquid welfare (the revenue op- timum), AAGBudget’s performance ranges from 94–99% of the optimum across inventory units, whereas AAGNoBudget ranges from 54–65% and Optimal SPA from 60–90%. Our algorithm’s revenue performance is consistently close to the liquid welfare benchmark across all values of r and outper- forms all other algorithms, which demonstrates that our algo- rithm has a stable and much better performance in practice than the theoretical 1 -approximation guarantee.