Choosing an appropriate model for survival analysis with recurrent events and competing risks I’m having some difficulties in selecting the appropriate model(s) for analyzing survival data which includes recurrent events and (potentially) competing risks. There is no particular ‘terminal’ event in the study. The dataset describes groups (dyads) in which 2 subjects play with each other for a defined period of time. During their gameplay, each subject can make a move without permission of the other player (call it a “p-move”). Each p-move is considered an event. The unit of analysis is the dyad. The analysis includes time-varying covariates.
Issue1: the first p-move (likelihood and time-to-event)
Since the p-move can be made by either player, would it be correct to use a competing risks model where p-move by player1 would be event1 and p-move by player2 would be event2? If so, I’m concerned with how to differentiate between the two players. If for example I use “relative experience” to define the players in each dyad, so that a p-move by the more experienced player (within the dyad) would be event1 and the less-experienced player would be event2, how do I treat censored dyads where no p-moves are made? Is it ok to define non-events as “0” even though there is no differentiation between the players in these dyads?
Issue2: repeated p-moves
Within each dyad, multiple p-moves can be made during the defined time period of the gameplay (all are the same type of move). There are several scenarios within each dyad: (1) single p-move is made by one player, (2) two p-moves are made, one by each player, (3) several p-moves are made by one of the players at different times, (4) several p-moves are made by one of the players at different times and then one or several p-moves are made by the other player, (5) ‘ping-pong’ p-moves are made by the players (e.g., player1, player1, player2, player1, player2, player2...), (6) no moves are made. Example (simplified):

Columns: ‘id’ is the dyad-id, ‘event’ is a binary variable for any p-move (by either player), ’order’ is the order of p-moves, ‘type’ is the initiating player at each point, ‘event2’ is an event-type (0,1,2) per player (though, this variable would need to be defined according to the differentiating characteristic of the player within the dyad, such as “relative experience”. This example is just for illustration)
Interest-questions: I would like to examine (a) the likelihood of each scenario based on dyad-characteristics, and player-characteristics, (b) whether the timing (‘tstop’) of the first p-move affects the likelihood of subsequent p-moves (i.e., scenarios 2,3,4,5), and their timing, (c) whether certain initial conditions (e.g., type of gameplay) affect the first p-move more strongly than it affects subsequent p-moves.
Questions about setup and modelling (Issue 2)

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*Is it correct to say that for most of what I am looking to examine (in Issue 2), I should not include the censored dyads (as in scenario 6) but use the subset dyads with at least 1 p-move? Since, for (a) I have no way of differentiating player-characteristics that are not ‘relative’ within the dyad (for example absolute measure of “experience” regardless of the other player), and for (b) there must be a first move.

*How do I model event type (competing risks(?)) with repeated events? If, for instance, the entire sample is made of scenarios 1 (single p-move) and 3 (several p-moves by one player), I can maybe use conditional PWP models (with or without frailty), and using column ‘order’ of events. What is the appropriate model for examining scenarios 4 and 5 where p-moves are made by both players? Would it be correct to use the same method with an added covariate for the initiating-player (such as column ‘event2’)?

Thank you.

Whether you take Frank Harrell's advice to handle this via a state-transition model or use a survival model, you still need to define the events/transitions carefully. Some thoughts on that.
First, do not omit censored data. If neither player chooses to make a move during a game, that contains information.
Second, you ask:

would it be correct to use a competing risks model where p-move by player1 would be event1 and p-move by player2 would be event2? If so, I’m concerned with how to differentiate between the two players.

That makes sense. A single event indicator with 3 levels--censored, Player1move, Player2move--would seem to be simplest. The tstart, tstop, eventType counting-process data format can incorporate any combinations of repeated events of different types, where eventType represents the status at tstop.
In each game you arbitrarily call 1 individual "Player_1" and the other "Player_2"; the order doesn't matter. You just include separate individual-specific covariates (e.g., number of prior games) as correspondingly labeled covariates. That lets you model associations of Player_1 and Player_2 characteristics separately with respect to Player1move and Player2move events.  To avoid confusion (and simplify later collection of results by individual), I'd use ID to represent individual players and a separate game indicator for labeling games/dyads.
Third, you need to give careful thought to how to incorporate within-game history into the model. You might, for example, need to include things like the numbers of prior Player1move and Player2move events, or whether the immediately prior event was by Player_1 and Player_2, as time-varying covariates. In principle, you could use extensions of that approach to model any of you scenarios.
Fourth, it might make the most sense to use a formal multi-state model as Frank Harrell suggests. For example, you could label states with respect to the number of moves by (Player_1, Player_2) in order: (0,0) can transition to (1,0) or (0,1); (1,0) can transition to (1,1) or (2,0), etc. That incorporates total and individual-specific prior numbers of events directly into the structure of the model and can let you focus more on other covariates of interest.
