Prove that, in a vickrey (second-price sealed-bid) auction, for any false bid bi by any single bidder (that is, if vi is the bidder's private value for the item, then bi≠vi), there exists a vector b−i of bids by all other bidders such that the false bid gives strictly worse utility than bidding the true valuation, that is, ui(bi, b−i)

cognitive overload: having a lot on your mind impairs decision-making and tends to result in the simplest, but not necessarily best, option being selected.

empathy gaps: overlooking how you might feel in a different situation can result in unnecessary purchases, such as overbuying when shopping for food on an empty stomach.

optimism and overconfidence: wearing rose-tinted glasses and having unrealistic expectations about the future can affect money management and leave you unprepared for a change in circumstance.

instant gratification: seeking instant gratification drives impulsive spending and can undermine long-term planning and savings.

harmful habits: automatic or mindless behaviour can amplify a poor financial decision as it becomes a recurring event.

social norms: we are heavily influenced by the actions of others; while this can be in certain circumstances, it also contributes to the pressure to keep up with the joneses through conspicuous consumption.

answer(b); ; ; the quantitative or qualitative variable for which the experimenter wishes to determine how its value is affected by the explanatory variable;

answercountries with higher real income per capita tend to have higher levels of this index.