Techniques are described for facilitating interactions between computing systems, such as by using an authorization system to automatically authorize financial payments between parties in accordance with previously specified private authorization instructions of at least one of the parties. In some situations, some or all of the payments are associated with commerce-related or other transactions, such as transactions initiated by a consumer via the Web to acquire items from a retailer. The authorization instructions may include predefined instruction sets that regulate conditions under which a potential payment can be authorized, with the instruction sets each associated in some situations with a reference. After one or more parties each supply one or more such references or otherwise indicate one or more such instruction sets for use with a potential payment, the authorization system can determine whether to authorize the payment based on whether the instruction sets are compatible or otherwise satisfied.