A four-party model provides a comprehensive set of interlocking legal contracts that detail roles, responsibilities, and technical methods. In order to take part in the network, each party must agree to one of the contracts in a given framework. Identity providers specialize in providing support for particular roles.
Examples: The credit card networks, such as Visa and Mastercard, are implemented as four party networks. These represent a large collection of individuals and institutions, each of which must routinely trust participants they’ve never encountered before.
Parties of all types continually join and leave the network, making it impractical for any single organization to track them all. By creating a standard set of well defined roles that work together, the Visa and Mastercard enable risk assessors to specialize.
Because of the vast difference in the size of the entities involved (anywhere from an individual person to a multi-national corporation), and the complexity of governing law, no single contract could be both complete and understandable by all parties.
To solve this problem, the network created a comprehensive, interlocking set of contracts that lay out all of the roles that entities can play. For each role, the appropriate contract specifies the interactions and responsibilities. The network design allows for multiple identity providers, each of whom can specialize in managing risk for a particular set of users. Risks are managed at the system level.
When to use: Closed network where all parties can be expected to sign a contract to join.
Advantages: Enables a network where participants of different sizes can interact smoothly with one another. Allows for specialization of risk management in a complex, constantly changing network where participants frequently join and leave.
Disadvantages: Depends upon the ability to create comprehensive contracts. Risk management can impose substantial costs on the network.
Ability to scale: Four party models can scale to a large number of participants.
The full papers is downloadable [Field-Guide-Internet-TrustID] Here is a link to introduction of the paper and a at the bottom of that post is a link to all the other models with descriptions. Below are links to all the different models.