Mathematical Framework

This section formalizes the reward and slashing functions, the settlement logic and presents a high-level simulation for parameter selection.

4.1 Performance Metric#

Let $P_i(t)$ denote the portfolio value of trader $i$ 's strategy at time $t$ , as reported by the oracle.

Define the periodic return over $[t,\; t+\Delta]$ as:

r_i(t,\Delta) = \frac{P_i(t+\Delta) - P_i(t)}{P_i(t)}

We assume the protocol operates in discrete settlement periods of length $\Delta$ (e.g., one week or one month).

Let $r_i$ denote the return over the current period and $\bar{r}_i$ the benchmark return (possibly zero or an index).

Define the excess return:

\tilde{r}_i = r_i - \bar{r}_i

4.2 Reward and Penalty Function#

The amount of LAMA transferred between the trader and the long/short pools depends on $\tilde{r}_i$ .

Let:

$B_i$ — the trader's staked bond
$\alpha$ — scaling coefficient capturing the maximum payout rate
$\beta$ — slashing coefficient

The protocol uses a piecewise linear function:

\text{Reward}_i = \max(0,\; \alpha \cdot \tilde{r}_i) \times B_i

\text{Slash}_i = \max(0,\; -\beta \cdot \tilde{r}_i) \times B_i

Thus positive excess returns trigger a reward equal to $\alpha \tilde{r}_i B_i$ (distributed to the long pool), while negative excess returns result in a slash of $\beta |\tilde{r}_i| B_i$ (distributed to the short pool).

:::info Parameter Design The coefficients $\alpha$ and $\beta$ can be calibrated to balance reward potential and risk. Typically $\alpha \leq \beta$ to discourage reckless risk-taking. :::

4.3 Settlement Function for Investors#

For each agent $i$ , define the long pool size $L_i$ and short pool size $S_i$ before settlement.

When $\tilde{r}_i > 0$ (agent outperforms):

Total reward $R_i = \alpha \tilde{r}_i B_i$ is added to the long pool
Amount $R_i$ is deducted from the short pool (pro rata from investors)

When $\tilde{r}_i < 0$ (agent underperforms):

Penalty $P_i = \beta |\tilde{r}_i| B_i$ is transferred from the trader's stake to the short pool
Same amount is deducted from the long pool

After settlement the net pool sizes become:

L_i' = L_i + \mathbb{1}_{\{\tilde{r}_i>0\}} R_i - \mathbb{1}_{\{\tilde{r}_i<0\}} P_i

S_i' = S_i - \mathbb{1}_{\{\tilde{r}_i>0\}} R_i + \mathbb{1}_{\{\tilde{r}_i<0\}} P_i

Where $\mathbb{1}$ is the indicator function.

Investor rewards (or losses) are proportional to their share of the pool before settlement.

4.4 Slashing Simulation Model#

To understand how different parameters affect the likelihood of slashing and reward magnitude, a simulation can be run on synthetic return paths.

Consider the return process $r_i(t, \Delta)$ modelled as a random variable with mean $\mu$ and standard deviation $\sigma$ .

One can simulate thousands of return paths and compute the expected value and variance of the payoff function:

\Pi_i(\tilde{r}_i) = \alpha \max(0,\; \tilde{r}_i) \cdot B_i - \beta \max(0,\; -\tilde{r}_i) \cdot B_i

Key outputs of the simulation:

Output	Description
$\Pr(\Pi_i < 0)$	Probability that the trader's stake is slashed
$\mathbb{E}[\Pi_i]$	Expected reward/penalty for a given $(\alpha, \beta)$
Distribution of outcomes	For long and short investors

By varying $\alpha$ and $\beta$ one can calibrate the system to produce desired risk-reward profiles.

:::tip Default Parameters A typical choice is $\alpha = 0.5$ and $\beta = 1.0$ , meaning traders receive half their excess returns while losing their stake at a 1

ratio when underperforming. :::

4.5 Numerical Example#

Suppose:

Trader stakes $B_i = 1{,}000$ LAMA
Settlement period $\Delta = 30$ days
Benchmark $\bar{r}_i = 0$
Parameters: $\alpha = 0.5$ , $\beta = 1.0$

Scenario A: Agent returns +4%

\tilde{r}_i = 0.04, \quad \text{Reward} = 0.5 \times 0.04 \times 1{,}000 = 20 \text{ LAMA}

20 LAMA is paid to long investors; the short pool loses 20 LAMA.

Scenario B: Agent returns −2%

\tilde{r}_i = -0.02, \quad \text{Penalty} = 1.0 \times 0.02 \times 1{,}000 = 20 \text{ LAMA}

20 LAMA is paid to short investors and deducted from the long pool.

:::danger Asymmetric Risk Note that a +4% return generates the same payout magnitude as a −2% return due to the asymmetric $\alpha / \beta$ ratio. This deliberately penalizes losses more heavily than it rewards gains, promoting conservative strategy design. :::