$TIDE Seashell Formulas

This document describes the generalistic formulas to allocate Seashells (Points) to individual users of TideFlow participating in free games and wager games. Seashells will later be converted into $TIDE tokens.

Participation in free and wager games is tracked separately

Silver Seashells
- Awarded for free games
- Reward currency for social tasks
Gold Seashells
- Exclusively earned for playing wager games

This ensures an effective airdrop allocation to players. Players of wager games will be adequately rewarded, preventing dilution by free-to-play users.

Determining Seashells Per Game

Initially we have to define the total number of seashells available for distribution in a single multiplayer game.

At its simplest, the total number of seashells within a single lobby is calculated as such:

$\text{Total Seashells in game =Total number of players General Boost Community Boost Wager Boost}$

General Boost: Note that the “General Boost” is set at 100% across all community sizes. This is in order to give two Seashells/player as a base rate.

Community Boost: Scale with the number of players that join one single game to incentivize larger communities.

Wager Boost (only in wager games): Scales with the wager each participant in a game bets by that incentivizing communities or games with larger wagers.

This simplified logic can further be expressed in the following way.

Let:

SH_{Total} = \text{Total amount of seashells available for distribution in a single (multiplayer) game}

N = \text{Total number of players in a single (multiplayer) game}

Boost_{b} = \text{The individual boost to incentivize large multiplier games according to the following table}

SH_{Total} = f(N) = N * \left( 1 + \displaystyle\sum_{b=0}^{B} Boost_b \right)

In a simplified example where only a base boost as well as a community boost is applied is shown in the following table:

Or visually:

Further expanding the above formula with a wager boost lead to a slightly more complicated table:

N	Base Boost	Large Game Boost	$SH_{Total}$
1	100%	0%	2
10	100%	0%	20
25	100%	0%	50
50	100%	0%	100
100	100%	100%	300
150	100%	100%	450
200	100%	100%	600
250	100%	200%	1000
500	100%	200%	2000
1000	100%	500%	7000

Or visually:

Further expanding the above formula with a wager boost lead to a slightly more complicated table:

N	Base Boost	Large Game Boost	Wager	Wager Boost
1	100%	0%	1	0%
10	100%	0%	10	0%
25	100%	0%	25	0%
50	100%	0%	50	0%
100	100%	100%	100	100%
150	100%	100%	150	100%
200	100%	100%	200	100%
250	100%	200%	250	200%
500	100%	200%	500	200%
1000	100%	500%	1000	500%

This again leads to a two dimensional grid of total seashells to be distributed in a single game:

N / Wager	1	10	25	50	100	150	200	250	500	1000
1	2	2	2	2	3	3	3	4	4	7
10	20	20	20	20	30	30	30	40	40	70
25	50	50	50	50	75	75	75	100	100	175
50	100	100	100	100	150	150	150	200	200	350
100	300	300	300	300	400	400	400	500	500	800
150	450	450	450	450	600	600	600	750	750	1200
200	600	600	600	600	800	800	800	1000	1000	1600
250	1000	1000	1000	1000	1250	1250	1250	1500	1500	2250
500	2000	2000	2000	2000	2500	2500	2500	3000	3000	4500
1000	7000	7000	7000	7000	8000	8000	8000	9000	9000	12000

We can put the above also into a graphical representation:

Calculating Seashells Per Player

Now that the total amount of seashells available for distribution per game has been defined, we can break it down to a single player.

Let:

N = \text{total number of players in a single (multiplayer) game}

n_{i} = \text{the individual single player in a single (multiplayer) game}

Therefore:

N = \displaystyle\sum_{n_i=1}^{N} 1

To determine the number of seashells distributed to an individual single player we need to rank the player's performance against the other players.

Let:

Rank_{i} = \text{Rank of an individual player among all other players in the game}

The ranking has to be based on a metric which we will call:

KPI_{i} = \text{Performance of an individual player i among all other players in the game}

Where KPI may be based on a Success Score or Return Measure (ROI) suitable for ranking the players individual performance.

Each player is then ranked based on the KPI relative to all other participants in the match. The following illustrates such a distribution:

The individual performance will then be rewarded by boosting seashells available for distribution for players that scored in the top percentiles while the majority (the belly of the distribution) will get a small reward and the bottom tail may or may not receive a marginal reward.

This increase is currently set to always incentivize players to compete in order to climb as far up the P&L ladder as possible. An idle playstyle, where aiming for the middle of the leaderboard is rendered inefficient from an economic and pointnomic perspective

Taking above into consideration one can easily draft a rewards table assuming a game with $SH_{Total} = 1000$

Comment	Percentile	Seashell Allocation	Seashells
Maximum	100%	30%	300
	95%	25%	250
	90%	20%	200
	75%	10%	100
Median	50%	5%	50
	25%	4%	40
	10%	3%	30
	5%	2%	20
Minimum	0%	1%	10

Or visually:

The number of seashells available for distribution in a specific percentile then needs to be shared among all players that scored in the same percentile.

Furthermore, if there should be no player scoring in a particular percentile the seashells allocated to that percentile are held in an overflow. This overflow is eventually proportionally distributed relative to the base seashell allocation, to the percentiles with players. By doing so one ensures that players in higher percentiles always receive more than those in lower percentiles.

Comment	Percentile	Score	Allocation	Seashells	Number of Players in Percentile	Actual Distribution	Actual Distribution per Player in Percentile
Maximum	100%	16	30%	90	1	120	120
	95%	15.45	25%	75	0	0	0
	90%	14.9	20%	60	1	80	80
	75%	13.25	10%	30	5	40	8
Median	50%	10.5	5%	15	38	20	0.53
	25%	7.75	4%	12	44	16	0.36
	10%	6.1	3%	9	7	12	1.71
	5%	5.55	2%	6	2	8	4
Minimum	0%	5	1%	3	2	4	2
				Overflow	75	TRUE
				Nominator	225

One can now easily define the generalistic formula for the individual user.

Let

P_{p} = p \text{ Percentile e.g. } P_{10\%} = 10 \text{\% Percentile}

N_{p} = \text{The number of players scoring in a specific percentile } p

O = Overflow = {\displaystyle\sum_{p}^{P}} \text{ Seashells allocated to a percentile p where } N_{p} = 0

NOM = Nominator = {\displaystyle\sum_{p}^{P}} \text{ Seashells allocated to a percentile p where } N_{p} \neq 0

SH_{p} = \text{Basis Seashells allocated to a percentile } p = Allocation * SH_{Total}

SHA_{p} = \text{Actual Seashells allocated to a percentile } p = SH_{p} + O * \frac{SH_p}{NOM}

Therefore, the Seashells allocated to an individual player i scoring in percentile p is:

D_{p,i} = \frac{SHA_p}{N_p}