Suppose there are two bidders $i=1,2$ who may either have a low valuation $V_L$ or a high valuation $V_H.$ Bidders do not know others' valuations. In a first-price auction, where ties are broken by coin-flip, what is the the minimum bids (i.e. the lowest bid in the support $\underline{b_i}$ in $[\underline{b_i},\overline{b_i}]$ for $i=1,2$) for the $V_L$ type bidder?
In these auctions, strategies are some probability over the domain $[\underline{b_i},\overline{b_i}]$ (i.e. a mixed strategy). My guess is that the support for both the $V_L$ type player is $0$, i.e. that $$[\underline{b_i},\overline{b_i}]=(0,V_L].$$ Furthermore, the probability of bidding close to $0$ is quite trivial, since the bidder's chance of winning is also close to $0$. However, my lecturer argues that the support $\underline{b_i}=V_L$ for both players, but this would not make bidding worthwhile for the $V_L$ type, since she would receive $0$ surplus.