Add docs for combinatorial tokens (#90)

* Add docs for combinatorial tokens

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* Fix compiler error by removing search bar and localization

* Fix package/yarn

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* Update docs/learn/combinatorial-tokens.md

Co-authored-by: Chralt <chralt.developer@gmail.com>

* Update docs/learn/combinatorial-tokens.md

Co-authored-by: Chralt <chralt.developer@gmail.com>

* Update docs/learn/combinatorial-tokens.md

Co-authored-by: Chralt <chralt.developer@gmail.com>

* Update docs/learn/combinatorial-tokens.md

Co-authored-by: Chralt <chralt.developer@gmail.com>

* Fix typo

* Update docs/learn/combinatorial-tokens.md

Co-authored-by: Chralt <chralt.developer@gmail.com>

* Update docs/learn/combinatorial-tokens.md

Co-authored-by: Chralt <chralt.developer@gmail.com>

* Update docs/learn/combinatorial-tokens.md

Co-authored-by: Chralt <chralt.developer@gmail.com>

* Update docs/learn/combinatorial-tokens.md

Co-authored-by: Chralt <chralt.developer@gmail.com>

* Fix formatting

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Co-authored-by: Chralt <chralt.developer@gmail.com>

docs/learn/combinatorial-tokens.md

@@ -0,0 +1,161 @@
+# Combinatorial Betting
+
+Zeitgeist's combinatorial tokens allow users to combine outcome tokens of
+several markets and trade these combined tokens in a single liquidity pool.
+
+Much of Zeitgeist's implementation of combinatorial tokens is based on pioneer
+work done by Gnosis
+[here (documentation)](https://docs.gnosis.io/conditionaltokens/) and
+[here (implementation)](https://github.com/gnosis/conditional-tokens-contracts).
+
+## Collection and Positions
+
+Given a market with outcomes `A`, `B` and `C`, we can form _collections_ of
+these outcomes. A collection is notated as, for example, `A|B`, and specifies a
+bet that `A` _or_ `B` will happen.
+
+A _position_ is a collection combined with a collateral token. Currently, the
+type of collateral is still tied to markets on Zeitgeist, but that might change
+in the future and you'll be allowed to use any collateral with any market.
+
+Positions are used to actually make bets on more complicated claims about the
+future.
+
+## Splits and Merges
+
+Suppose we have two markets with outcome tokens `A`, `B`, `C` and `X`, `Y`, `Z`,
+resp. Their tokens can be combined to the following tokens: `A&X`, `A&Y`, `A&Z`,
+`B&X`, `B&Y`, `B&Z`, `C&X`, `C&Y`, `C&Z`. A bet on the token `A&X` is a bet that
+both `A` _and_ `X` occur (we'll interpret the meaning of `A&X` in case one or
+both markets is a scalar market below in the section _Redeeming_).
+
+In fact, we can even go further and combine positions from both markets into
+_split positions_ such as `(A|B)&Z`, which signifies that `A` or `B` and `Z`
+will occur.
+
+Split collections are obtained by taking a position such as `(A|B)` of one of
+the markets and specifying a partition of the outcomes of the other market, such
+as `(X|Y)` and `Z` (the partition must cover all outcomes and there may be no
+duplicates). Splitting `x` units of `(A|B)` using the partition `(X|Y), Z`
+results in `x` units of the following tokens: `(A|B)&(X|Y)` and `(A|B)&Z`.
+
+Splitting can even involve more complicated tokens. For example, the position
+`(A|B)&(X|Y)` may further be split into `(A|B)&(X|Y)&U` and `(A|B)&(X|Y)&V` if
+there is a market with outcomes `U`, `V`.
+
+Merging is the opposite process. For example, `x` units of `(A|B)&(X|Y)` and
+`(A|B)&Z` may be combined into `x` units of `(A|B)`.
+
+## Redeeming
+
+If a user owns a position involving multiple markets, such as `(A|B)&X` and only
+one of the markets is resolved (say, the market with `A`, `B`, `C`), then they
+may want to cash out the part of the position. This is where payout vectors come
+in.
+
+The _payout vector_ of a resolved prediction market determines the value of each
+outcome token of that market. For a market with outcome tokens
+$x_1, \ldots, x_n$, the payout vector $(p_1, \ldots, p_n)$ specifies how much
+collateral one unit of each outcome token redeems for.
+
+For example, if a categorical market with three outcomes resolves to the second
+outcome, the payout vector is $(0, 1, 0)$. If a scalar market with range
+$[1, 3]$ resolves to $2.5$, then the payout vector is $(.75, .25)$ assuming that
+the first outcome is _Long_.
+
+The payout of a collection like `(A|B)` is the sum of the payouts of each
+individual outcome token. Suppose that the payout vector for the market with
+`A`, `B`, `C` is `(0, 1, 0)`, then `(A|B)` has a payout of `1`.
+
+Returning to the idea of redeeming tokens, `x` units of `(A|B)&X` can be
+redeemed for `x` units of `X` since the payout of `(A|B)` is equal to `1`. To
+take a more complex example, if we have a scalar market with tokens `L` and `S`
+(long and short) and a payout vector of `(.6, .4)` and we want to redeem `x`
+units of `L&(X|Y)`, then we receive `.6` times `x` units of `(X|Y)`.
+
+## Combinatorial Betting
+
+Buying individual outcomes of a combinatorial market allows the user to bet on
+more specialized outcomes (A _and_ B will happen), but the true strength of
+combinatorial markets lies in making bets on contingencies (if A occurs, then so
+will B).
+
+The following example is taken from [H13]. Let $D$ denote the event that the
+Democratic Party wins the 2016 U.S. Presidential election and let $H$ denote the
+event that Hillary Clinton is nominated as the Dem's candidate. The pairs of
+securities $(D, \neg D)$ (i.e. "Pays 1\$ if $D$" and "Pays 1\$ if not $D$") and
+$(H, \neg H)$ are traded on separate markets, but can be combined into a single
+market by using _splits_.
+
+If you know how buying complete sets works on Zeitgeist, you already know one
+example of a split: The collateral token is split into a set of tokens which,
+when the markets resolves, will have the same value as the collateral token. By
+the same reasoning fashion, we can split non-collateral tokens like "Pays 1\$ if
+$D$" into "Pays 1\$ if $D$ and $H$" and "Pays 1\$ if $D$ and not $H$". For the
+sake of simplicity, we denote these tokens by $D$, $D \land H$ and
+$D \land \bar H$, resp. By combining the two markets above, we create a new
+market with four assets: $D \land H$, $D \land \bar H$, $\bar D \land H$ and
+$\bar D \land \bar H$.
+
+Using splits, agents can now bet on correlations between the events $D$ and $H$.
+For example, to bet that the Dems win if Hillary is nominated, the agent would
+split their collateral into $x$ units of $H$ and $\bar H$, then split these
+units into $x$ units of $H \land D$ and $H \land \bar D$, then sell their units
+of $H \land \bar D$ to acquire more $H \land D$. There are three possible
+outcomes:
+
+- Hillary is nominated and wins the election. Each unit of $H \land D$ pays 1\$.
+  The agent makes a profit as they own more than $x$ units.
+- Hillary is nominated and loses the election. All assets that the agent holds
+  are worthless.
+- Hillary is not nominated. Each unit of $\bar H$ pays 1\$. The agent receives
+  their buy-in of $x$ back (the bet is essentially declared void).
+
+The spot price of this swap is equal to the
+[conditional probability](https://en.wikipedia.org/wiki/Conditional_probability)
+$P(D|H)$ predicted by the market. We denote the trade described above by
+$H \Rightarrow D$.
+
+To prevent arbitrage opportunities, buying such a composite asset must not move
+the price of uninvolved outcomes. For example, betting that $D$ holds if $H$
+holds should leave the price of $\bar H$ unchanged. After all, the agent is
+betting on what happens contingent on $H$; they're not making any claims as to
+how likely $H$ is to occur. This is summarized in the _modularity property_
+defined by Hanson in [H03a]:
+
+> Consider the case of an agent who bets with a market scoring rule, and bets
+> only on the event $A$ given the event $B$. That is, this agent gains assets of
+> the form “Pays \$1 if $A$ and $B$ hold”, in trade for assets of the form “Pays
+> \$1 if $B$ holds and $A$ does not. [...] In general, depending on the
+> particular market scoring rule, such a bet might change any probability
+> estimate $p_i$, and thus change any event probability
+> $p(C) = \sum_{i \in C} p_i$. It seems preferable, however, for this bet to
+> change as little as possible besides $p(A|B)$ (and of course
+> $p(\bar A|B) = 1 − p(A|B)$).
+
+(The market maker implemented in zrml-neo-swaps has that property.)
+
+A fundamental idea is that trades made on prediction markets are not so much
+absolute bets ("I bet $10 that Hillary will win at 1:1 odds"), but rather that
+trades are made to "correct" the forecast of the market by changing the prices
+of the outcomes. The agent executing the trade pays a price to do so, but is
+rewarded if the prediction is closer to the truth than the current one.
+
+One could say that the agent buying $H \Rightarrow D$ claims that $H \land D$ is
+undervalued and $H \land \bar D$ is overvalued, and that $\bar H \land D$ and
+$\bar H \land \bar D$ are correctly priced or the agent doesn't know how to
+profitably change their price.
+
+Viewed from this angle, a _combinatorial bet_ divides the assets in a
+combinatorial pool into three categories: _buy_ (the assets the agent thinks are
+undervalued), _sell_ (the assets the agent thinks are overvalued) and _keep_
+(the assets whose prices the agent can't or won't change).
+
+## Bibliography
+
+- [H03a] R. Hanson, "Logarithmic Market Scoring Rules for Modular Combinatorial
+  Information Aggregation," The Journal of Prediction Markets, vol. 1, no.
+  1, 2003. [Online]. Available:
+  [https://doi.org/10.5750/jpm.v1i1.417](https://doi.org/10.5750/jpm.v1i1.417)
+- [H13] R. Hanson, "Shall We Vote on Values, But Bet on Beliefs?," The Journal
+  of Political Philosophy, vol. 21, no. 2, pp. 151-178, 2013.

docs/learn/using-zeitgeist-markets.md

@@ -290,52 +290,15 @@ as he owns no more "Yes".
 
 ### The Liquidity Pool
 
-Zeitgeist supports a constant mean market maker, also referred to as constant
-product market maker or CPMM on our platform, which is based on the
-[Balancer AMM](https://balancer.fi/whitepaper.pdf), a variation on the basic
-$x \cdot y = \mathrm{const}$ formula which allows different assets to have
-different _weights_, which define their impact on price.
-
-This AMM's liquidity pool contains balances of the base asset (e.g. ZTG) and of
-all outcome tokens of the market. Users can trade their tokens with tokens
-stored in the pool: They can buy outcome tokens from the pool with collateral
-which is then added to the pool, or sell outcome tokens to the pool for some of
-the pool's collateral.
+Zeitgeist supports a constant function market maker called DLMSR, which is based
+on Robin Hanson's DLMSR, a variation on the basic $x \cdot y = \mathrm{const}$
+formula.
 
 By default, a new market has no liquidity pool. Instead, the pool must either be
 deployed by the market creator, or by some external liquidity provider. After
 the pool is created, others may _join_ the liquidity pool by providing
-additional liquidity. When deploying liquidity into a pool, a liquidity provider
-will usually provide the same amount of complete sets of outcome tokens as
-collateral ($x$ of each outcome token and $x$ ZTG). The current minimum for $x$
-is $.1$ units of collateral, making a total value of $.2$ units of collateral.
-
-For example, lets say the JWST market has no liquidity pool yet and Alice wishes
-to deploy a pool. First she mints 100 complete sets of outcome tokens, so she
-pays 100 ZTG into the prize pool of the market and receives 100 "Yes" and 100
-"No". Then she transfers these outcome tokens plus 100 ZTG into the pool. The
-whole endeavor costs her 200 ZTG plus transaction costs and earns her 100
-liquidity shares.
-
-Suppose now that the market ends and the balances of the pool are the following:
-63 "Yes", 89 "No", and 120 ZTG. The balance of ZTG has increased from trading
-fees. After market close, Alice withdraws her funds: The outcome tokens and 120
-ZTG. If the JWST did not launch on December 18, then she can redeem the 89 "No"
-for 89 ZTG from the prize pool. This means that she's made a gain of 9 ZTG for
-supplying liquidity to the pool. If, on the other hand, the JWST does launch
-December 18, Alice is left holding 120 ZTG and 63 "Yes" (redeemable for 63 ZTG).
-As a result, Alice will incur a net loss of 17 ZTG, while many traders might
-realize a profit.
-
-For example, if the pool contains 100 ZTG and 100 "Yes" and Alice buys 3 "Yes"
-at 0.5 ZTG, then Alice transfers 1.5 ZTG into the pool and receives 3 "Yes" from
-the pool, leaving the pool with 101.5 ZTG and 97 "Yes" (ignoring transaction
-cost, trading fees and slippage).
-
-Let's pretend that the automated market maker has now adjusted the price of
-"Yes" to 0.6 ZTG and Bob wants to sell 5 "Yes". He would receive 3 ZTG from the
-pool and add 5 "Yes", leaving the pool at 98.5 ZTG and 102 "Yes" (ignoring
-transaction cost, trading fees and slippage).
+additional liquidity. For details on the function of the liquidity pool, see
+https://github.com/zeitgeistpm/zeitgeist/blob/main/zrml/neo-swaps/README.md.
 
 Note that this means that trading can only happen when the liquidity pool is
 sufficiently deep. If the pool is too shallow, some trades may be impossible

docusaurus.config.js

@@ -16,17 +16,6 @@ const config = {
   organizationName: "zeitgeistpm",
   projectName: "documentation", // Usually your repo name.
 
-  // support multi-languages
-  i18n: {
-    defaultLocale: "en",
-    locales: ["en", "zh-CN", "ru"],
-    localeConfigs: {
-      en: {
-        htmlLang: "en-GB",
-      },
-    },
-  },
-
   themeConfig:
     /** @type {import('@docusaurus/preset-classic').ThemeConfig} */
     ({

sidebars.js

@@ -29,6 +29,7 @@ const sidebars = {
         "learn/governance",
         "learn/court",
         "learn/order-book",
+        "learn/combinatorial-tokens",
         "learn/futarchy",
         "learn/comparisons",
         {