18
Let us now consider, however, that if all beliefs are zero (both players believe that the other is
small with probably one, and have zero belief that there are AF institutions in the target) then the
strategy TT weakly dominates all other strategies for both players. In this case, rational players
will be expected to end up at ({TT
1
; TT
2
}). This makes sense, given the argument. If players are
certain the other player has a (weakly) dominant strategy to take and there is no possibility of
encountering favorable institutions in the target, taking makes sense, regardless of your type.
We can reduce this 4x4 matrix by eliminating strategies that are weakly dominated (note
that the iterated elimination of weakly dominated strategies is not order-dependent (Duggan
2003)). For player 1, the first row (from the top - (TT
1
)) weakly dominates the second row (TL
1
)
and the third row (LT
1
) weakly dominates the bottom row (LL
1
). Similarly for player 2, the first
column (from the left - (TT
2
)) weakly dominates the second column (TL
2
) and the third column
(LT
2
) weakly dominates the fourth (LL
2
). This iterated elimination of weakly dominated
strategies results in the reduced 2x2 matrix of Figure 7 (for a discussion of iterated elimination of
weakly dominated strategies see (Brandenburger, Friedenberg, and Keisler 2004; Brandenburger
and Keisler 2000)).
Figure 7: Reduced Strategic Form
Player 2 column
Player 1 row TT LT
TT
0, 0
2γc, β
LT
γ, 2βc αγ+γc, αβ+βc
This accords with what we already know about small W types from the extensive form game,
any strategy which involves small types leaving is dominated.
We can also look for conditions on the beliefs that might produce more pure strategy
Nash equilibria. We can note that as long as each player’s belief that the rival is a large W type is
Convention