Revenue from the Saints, the Showoffs, and the Predators: Comparisons of Auctions with Price-Preference Values Supplemental Information
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1 Revenue from the Saints, the Showoffs, and the Predators: Comparisons of Auctions with Price-Preference Values Supplemental Information Timothy C. Salmon Florida State University R. Mark Isaac Florida State University March, 2004 The tables below contain all of the numerical results from the main paper that are either summarized in the figures or described in the text. We have also included results of significance tests between the revenue and efficiencies achieved for every set of parameters run. The tests on the revenue numbers are general t-tests and Wilcoxon rank sum tests with the numbers included representing the p-values of the test. For the efficiency comparisons, the tests are paired t-tests and signed rank sum tests, a paired version of the Wilcoxon test. The numbers also are the p-values of the test. Standard Charity Figure : Revenue comparison assuming uniformly distributed values between second price and first price auctions and between the standard and then baseline symmetric charity cases. Expected Revenue Second Price First Price α,β=0 α,β=.5 α,β=.3 α,β=0 α,β=.5 α,β=.3 α,β=0 α,β=.5 α,β=.3 N=2 N=4 N=6 Support for Figure :
2 Revenue Efficiency α i = (.20) β i = (.77) (.044) α i = (.280) β i = (.254) (.058) α i = (.380) β i = (.209) (.03).3477 (.2395).6028 (.202).755 (.590).4097 (.969).6588 (.770).7562 (.355).485 (.808).7040 (.5).7733 (.263) NA NA NA NA NA NA NA NA NA 2
3 Figure 2: Revenue comparison for the standard and baseline symmetric charity cases assuming normally distributed values Second Price First Price : 0 First Price : Expected Revenue α,β=0 α,β=.5 α,β=.3 α,β=0 α,β=.5 α,β=.3 α,β=0 α,β=.5 α,β=.3 N=2 N=4 N=6 Normal distribution, µ =.5, σ =.5 Support for figure 2: ᾱ = β = α i = β i Revenue Efficiency α i = (.0539) β i = (.0460) (.048) α i = (.0596) β i = (.058) (.0473) α i = (.0849) β i = (.0677) (.0594).483 (.2).5434 (.0898).600 (.079).460 (.0828).5600 (.0850).6030 (.0795).498 (.0729).5780 (.0746).6205 (.0779) (.0055) NA NA.374 NA NA NA NA NA NA 3
4 ᾱ = β =0 Revenue Efficiency α i = (.0546) β i = (.0450) (.047) α i = (.050) β i = (.0443) (.048).462 (.0982).5760 (.0863).69 (.0752).4857 (.0880).5975 (.0796).6342 (.0772) (.0095) (.0095) NA NA NA NA 4
5 2 See and Be Seen Figure 3: Revene comparisons for SBS model assuming uniformly distributed values. Expected Revenue Second Price First Price β={0,.5} β={0,.5} β={0,.5} β={0,.5} β={0,.5} β={0,.5} N=2 N=4 N=6 Support for figure 3: β =0 Revenue Efficiency β i = (.334) α i = (.432) (.275) β i = (.2468) α i = (.3054) (.2886).3525 (.2462).644 (.237).7734 (.763).423 (.279).864 (.3322).084 (.3450) (.0206) (.037) (.0330).94 (.264).970 (.376).93 (.307).9953 (.0206).9889 (.037).9866 (.0330).94 (.264).970 (.376).93 (.307) β =.5β i Revenue Efficiency β i = (.334) α i = (.398) (.224) β i = (.2390) α i = (.32) (.2996).3609 (.2467).6434 (.2096).7835 (.733).429 (.285).8598 (.3463).0897 (.3528) (.0243) (.0297) (.0353).9403 (.256).950 (.400).982 (.290).9937 (.0243).990 (.0297).985 (.0353).9403 (.256).950 (.400).982 (.290) 5
6 β = β i Revenue Efficiency β i = (.35) α i = (.408) (.368) β i = (.2434) α i = (.325) (.2987).3484 (.2533).6557 (.2233).772 (.767).4297 (.282).8504 (.3487).087 (.3593) (.0226) (.0292) (.0346).938 (.366).920 (.35).950 (.339).9945 (.0226).9900 (.0292).9864 (.0346).938 (.366).920 (.35).950 (.339) 6
7 Figure 4: Revenue comparison for the SBS model for normally distributed values. Expected Revenue Second Price First Price : 0 First Price :.5 First Price : β={0,.5} β={0,.5} β={0,.5} β={0,.5} β={0,.5} β={0,.5} N=2 N=4 N=6 Support for figure 4: β =0 Revenue Efficiency β i = (.0403) α i = (.0403) 6.66 (.0389) β i = (.0322) α i = (.0385) (.037).4469 (.323).5908 (.036).6555 (.086).4890 (.439).8542 (.2049).0066 (.897) (.0372).9835 (.038).9862 (.0327) (.30).899 (.246).976 (.009).9872 (.0340).9850 (.0352).9833 (.0368).9050 (.442).9022 (.362).900 (.22) β =.5β i Revenue Efficiency β i = (.0625) α i = (.0505) (.0477) β i = (.07) α i = (.0698) (.0595).4487 (.32).5968 (.0985).6536 (.0892).503 (.484).8580 (.965).9947 (.937) (.0302) (.036) (.0323).8963 (.439).9048 (.248).907 (.43).9896 (.030).983 (.0377).9825 (.0377).8859 (.529).9060 (.305).904 (.28)
8 β = β i Revenue Efficiency β i = (.0765) α i = (.063) (.0564) β i = (.67) α i = (.400) (.265).45 (.363).5854 (.0996).6580 (.0898).4879 (.492).858 (.2087).0000 (.857) (.0322).9859 (.0347) (.0363).8922 (.494).96 (.284).97 (.32).9887 (.036).9865 (.0338).987 (.0392).8953 (.462).952 (.237).950 (.6)
9 3 Raising Rivals Cost Figure 5: Revenue comparisons for RRC model assuming uniformly distributed values. Expected Revenue Standard Second Price : 0 Second Price :.5 Second Price : First Price α={0,.5} α={0,.5} α={0,.5} α={0,.5} α={0,.5} α={0,.5} N=2 N=4 N=6 ᾱ =0 Revenue Efficiency β i = (.44) α i = (.83) (.03) β i = (.04) α i = (.347) (.264).362 (.202).5854 (.725).6903 (.407).359 (.782).567 (.375).6508 (.20) (.0089) (.069) (.0246).990 (.0347) (.0659) (.0776).9894 (.078).9950 (.0287).9924 (.023).9707 (.59).98 (.0559).9757 (.0540) ᾱ =.5α i Revenue Efficiency β i = (.26) α i = (.90) (.074) β i = (.44) α i = (.369) (.255).3477 (.249).5804 (.707).6779 (.409).33 (.674).5443 (.429).6302 (.330) (.0070) (.057) (.0222) (.0306) (.0673).9627 (.0733).9926 (.0573).9959 (.079).9928 (.022).975 (.24).9728 (.0625).9692 (.0646)
10 ᾱ = α i Revenue Efficiency β i = (.63) α i = (.59) (.047) β i = (.) α i = (.287) (.33).3385 (.2036).5879 (.735).6786 (.408).390 (.580).589 (.470).6059 (.427) (.0063) (.053) (.022) (.0379) (.0646).9575 (.0786).9934 (.0553).9939 (.0239).997 (.0236).9763 (.0886).9665 (.0735).9574 (.0798)
11 Figure 6: Revenue comparisons for RRC model assuming normally distributed values. Expected Revenue Standard Second Price First Price : 0 First Price :.5 First Price : α={0,.5} α={0,.5} α={0,.5} α={0,.5} α={0,.5} α={0,.5} N=2 N=4 N=6 ᾱ =0 Revenue Efficiency β i = (.0586) α i = (.05) 6.57 (.0484) β i = (.0683) α i = (.063) (.0586).4055 (.05).5230 (.085).5682 (.0744).4004 (.0760).4963 (.0665).5352 (.0683) (.090) (.0253) (.0249) (.0743) (.0966).9374 (.090).9949 (.03).9947 (.082).995 (.0228).9793 (.0580).9656 (.0657).9564 (.076) ᾱ =.5α i Revenue Efficiency β i = (.053) α i = (.0446) (.0409) β i = (.0464) α i = (.049) (.036).4094 (.0879).4976 (.0652).5398 (.0700).3674 (.0724).47 (.0830).594 (.0832).9948 (.084) (.0246).9904 (.0244) (.0690) (.0830).9478 (.0807).9855 (.0435).9745 (.0566).9573 (.0720).9642 (.0738).949 (.0840).944 (.088)
12 ᾱ = α i Revenue Efficiency β i = (.0470) α i = (.0372) (.0356) β i = (.0326) α i = (.0273) (.0248).4095 (.0996).56 (.0862).566 (.0763).352 (.0754).4672 (.0872).5229 (.0853).9936 (.027).9900 (.0268).996 (.0226).967 (.0682).9425 (.0879).9445 (.087).993 (.0242).9920 (.0230).990 (.0240).9586 (.0796).9476 (.0853).9376 (.0922)
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