💻The final auction price calculated based on AMM

Example Calculation

Assume these auction parameters:

• Initial Pool: 10 ETH and 1,000,000 BID

• Total Participants: 30

• Price Multiplier: 4.9012 (calculated based on 30 participants)

• Participants bid on their desired amount of BID units. Prices adjust dynamically based on demand, and refunds are given if bids are insufficient.

Example Participant Information

Participant	Bid (ETH)	Units Requested
1	1.56	10
2	4.93	3

AMM Model Formula

The core of the AMM model is the constant product formula:

$$k=x\times y$$

where:

• $x=ETH\,in\,the\,pool$

• $y=BID\,in\,the\,pool$

• $k$ is a constant (initially set as the product of initial ETH and BID in the pool).

To calculate the ETH needed for a specific BID purchase:

1. Determine the new BID balance in the pool after the purchase:

$y\,{\tiny new}=y\,{\tiny new}-BID\,requested$

1. Use the constant kkk to find the new ETH balance:

$x\,{\tiny new}=\tfrac{k}{y\,{\tiny new}}$

1. Calculate the base ETH cost for the purchase:

$ETH\_needed\_base=x\,{\tiny new}-x\,{\tiny initial}$

Calculation Steps

1. Calculate Required BID Units and Base ETH Cost:

Participant 1 requests 100,000 BID (10 units at 10,000 BID each). The base ETH cost calculated using the AMM model is 1.11 ETH.

Participant 2 requests 30,000 BID (3 units). The AMM base cost for these units is 0.31 ETH.

1. Apply Price Multiplier:

Participant 1: 1.11 ETH * 4.9012 = 5.45 ETH

Participant 2: 0.31 ETH * 4.9012 = 1.51 ETH

1. Determine Auction Outcome:

Participant 1 bid 1.56 ETH, which is below the required 5.45 ETH, so they do not win and receive a full refund.

Participant 2 bid 4.93 ETH, which covers the 1.51 ETH needed, so they successfully purchased their 3 units. Their final cost is 1.51 ETH, and they receive a refund of 3.42 ETH.

Summary

Participant	Bid (ETH)	Units Requested	Successful Units	Cost (ETH)	Refund (ETH)
1	1.56	10	0	0.0	1.56
2	4.93	3	3	1.51	3.42

In this setup, the AMM model dynamically adjusts prices based on demand. Successful participants pay the AMM-adjusted price, while unsuccessful bids receive full refunds.

Examples of simulated auction results

• Total participants: 30

• Final price multiplier: 4.9012

• Total successful ETH spent: 9.8260 ETH

• Total successful BID units purchased: 167

• Average price per unit: 0.05883832 ETH

• Total successful bids: 28

Participants in the auction

Participant	Bid (ETH)	Units Requested	Successful Units	Cost (ETH)	Refund (ETH)
1	1.56	10	10	0.4951	1.0649
2	2.93	3	3	0.1505	2.7795
3	1.65	5	5	0.2528	1.3972
4	0.36	5	5	0.2554	0.1046
5	0.33	8	0	0.0	0.33
6	3.36	2	2	0.1029	3.2571
7	2.59	2	2	0.1033	2.4867
8	1.82	4	4	0.2079	1.6121
9	2.41	9	9	0.4742	1.9358
10	4.44	9	9	0.4832	3.9568
11	0.95	4	4	0.2177	0.7323
12	2.63	6	6	0.33	2.3
13	1.07	8	8	0.4466	0.6234
14	3.07	9	9	0.5117	2.5583
15	1.22	9	9	0.5217	0.6983
16	3.72	6	6	0.3536	3.3664
17	3.07	2	2	0.1189	2.9511
18	4.77	2	2	0.1194	4.6506
19	3.74	10	10	0.6051	3.1349
20	1.49	9	9	0.5563	0.9337
21	4.42	4	4	0.2509	4.1691
22	3.91	8	8	0.5086	3.4014
23	0.32	8	0	0.0	0.32
24	2.64	4	4	0.2578	2.3822
25	3.99	7	7	0.457	3.533
26	4.11	9	9	0.5985	3.5115
27	4.19	9	9	0.6113	3.5787
28	0.83	1	1	0.0687	0.7613
29	4.95	6	6	0.4158	4.5342
30	4.49	5	5	0.3511	4.1389