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While reading Pearce(1984), I can't understand what "agent" $j$ and agent $ij$ mean on page 1041:

Consequently I associate a conjecture $$ c^{ij} = (c^{ij}(1), ... , > c^{ij}(N)) $$ with each information set $I^{ij}$ in $\Gamma$; $c^{ij}(k)$ represents what an "agent" $j$ for player $i$ believes, once $I^{ij}$ is reached, about what player $k$'s mixed strategy is. A conjecture $c^{ij}(k)$ over a set $A^k \subseteq M^k$ can be regarded as an element of $\bar{A}^k$ (see Appendix A).
I have noted that an agent $ij$, upon being reached, should not entertain a conjecture that does not reach $I^{ij}$. A further restriction, not invoked in other solution concepts, is appropriate: if the information set can be reached without violating the rationality of any player, then the agent's conjecture must not attribute an irrational strategy to any player.

Some notations:
$I_{ij}$ is player $i$'s $j$th information set, $M_k$ is player $k$'s mixed strategy set and $\bar{A}^k$ is the convex cone of $A^k$.

So an agent is definitely different from a player, but what else can it be to form a belief?

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I have not read this paper in full, I am merely guessing based on some parts.

It seems to me that agent $ij$ is short for "agent $j$ for player $i$".

A player's strategy assigns a move/choice of action to each information set $I^{ij}$. Depending on the story of the game you can think about this move as being executed by the player herself or maybe just agents of the player. Kind of like with subgame perfection you can assume that the strategy must fullfil some conditions in each information set $I^{ij}$. When explaining these conditions it is easier to assume that the player does not have independent but somehow 'logically restricted beliefs' in every information set, but that all her agents have 'logically restricted beliefs'. So an agent $ij$ is basically just a player's strategy in $I^{ij}$.

By 'logically restricted beliefs' I mean the restrictions posed on conjectures.

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  • $\begingroup$ Is it the case that agent $j$'s objective coincide with player $i$? It seems to me Pearce(1984) introduced these hypothetical agents to circumvent the difficulty of constructing a single consistent belief system for player $i$, the arbitrariness of probability conditional on a null event, right? This difficulty is avoided by giving conditional probabilities first and deducing probabilities of events next in later paper. $\endgroup$ Jul 5 '15 at 7:42
  • $\begingroup$ Yes, agent $j$'s objective would coincide with player $i$. (You may think of the agents as split personalities of the player.) What you are saying about the motivation of Pearce makes sense but I cannot confirm it because unfortunately I have not read the paper in full. $\endgroup$
    – Giskard
    Jul 5 '15 at 8:44

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