Are there multiple equilibria in the second price auction?

Question

Suppose that $n\geq 2$ bidders compete in a second price auction. Each bidder $i$ knows their own valuation $v_i$, but only knows the distribution generating the valuations of the other players. Valuations are independently, continuously and symmetrically distributed. Define a (pure) strategy of a player as a function mapping from each valuation that they might have to a bid $b$; and define a (pure strategy) equilibrium as a set of (pure) strategies, one for each player, such that each player's strategy maximises their expected payoff given the strategies of the other players.

In this context, it is well known that it is weakly dominant for each player to bid their valuation, i.e. set $b(v_i) = v_i$. However, are there are other (pure strategy) equilibria of this game?

One point that perhaps should be clarified: in some discussion of second price auctions that I have seen, people define each player's strategy as their bid (a number) not their bidding functions. Here I am interested in the game obtained by defining strategies as functions.

Edit: in response to some helpful examples from @Giskard, we may wish to restrict attention to equilibria that (i) involve deviations from truthful bidding that occur with positive measure (ii) survive natural refinements that we might want to apply.

Answer 1

Sure. An example: if both valuations are drawn from the $[0,1]$ interval then the strategies $$ b_1(v_1) = v_1 $$ and $$ b_2(v_2) = \left\{\begin{array}{cc} v_2 & \text{ if } v_2 < 1 \\ 5 & \text{ if } v_2 = 1. \end{array}\right. $$ Another, slightly more annoying equilibrium for $v_1,v_2 \in [0,1]$: $$ \begin{align*} b_1(v_1) & = 0 \\ b_2(v_2) & = 2. \end{align*} $$

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EDIT: The truth-telling bidding strategies are weakly dominant strategies. If we only allow trembling hand perfect equilibria, players assign positive probabilities (in this case densities) to all truncated strategy profiles. In such a setting, the weakly dominant strategies become strictly dominant. (Because of sequential rationality even zero measure deviations such as my first example are eliminated.) In a game where all players have strictly dominant strategies, there is only one equilibrium.