Game theory run backwards: fix the rules so that honesty is the best move. The engine is the second-price (Vickrey) sealed-bid auction — highest bidder wins and pays the second-highest bid. Truth-telling becomes a dominant strategy, proven exactly over a grid of every deviation.
source W. Vickrey, Counterspeculation, Auctions, and Competitive Sealed Tenders, Journal of Finance 16(1), 1961, pp. 8–37 · doi.org/10.1111/j.1540-6261.1961.tb02789.x · rendered, not quoted.
Each bidder k holds a private value vk and submits a sealed bid bk. The mechanism is a pair of functions on the bid vector:
winner = argmaxk bk
price = maxj≠winner bj (2nd-highest bid)
Utility is uk = vk − price if k wins, else 0. Because the price a winner pays depends only on others' bids, no bidder can lower their own price by shading — so reporting vk truthfully is a dominant strategy.
Neighbour sphere: THE NASH EQUILIBRIUM — designed on purpose. Nash asks "given the rules, where do rational agents settle?" Mechanism design inverts it: choose the payoffs so the equilibrium is truth-telling.
Vickrey does better than a Nash equilibrium: truthful bidding is dominant (best whatever others do), not merely a mutual best-response. Strategyproofness — the corner of game theory that alignment and governance build on.
Live re-check of the running engine: it searches every bidder, value and deviation on the grid for a profitable underbid. Truthful ⇒ none exists.
Flip the payment rule in window 6 and this witness turns red within one render.
Three bidders with fixed private values. Bidders 2 & 3 bid truthfully; you drive bidder 1's bid to test whether deviating ever helps.
true values v = (7, 4, 2) · truthful bid = 7
Live outcome under the current rule (second-price):
—
Proven result — winner is the highest-value bidder, price is the second-highest value, and truth-telling is dominant:
The adversary's real leverage is elsewhere: collusion (rings agreeing to suppress bids) and a corrupt/uncommitted auctioneer who peeks at bids or invents a phantom second bid. Vickrey is truthful only when the seller is bound to the rule.
"Second-price just means the winner pays less, so sellers lose money."
Correction: revenue equivalence — in expectation, first- and second-price yield the same revenue with symmetric bidders; Vickrey buys truthfulness, not lower revenue.
"Truthful ⇒ also efficient in revenue for the seller."
Correction: Vickrey maximises allocative efficiency, not seller revenue. amber The revenue-maximising design (Myerson 1981) sets reserve prices and generally is not the plain Vickrey rule.
"Dominant-strategy truthful ⇒ safe to run in practice."
Correction: breaks under budget constraints, common values (winner's curse), and multi-unit settings.
Disclosed planted void: switch the rule to charge the winner their own bid (turn second-price into first-price). Truthful bidding is no longer dominant — the witness in window 7 catches a profitable underbid.
Under first-price a winner with value 5 facing a bid of 2 does strictly better bidding 3 (u = 2) than bidding 5 (u = 0).