You'll Never Guess the Optimal Strategy

Consider a three player game where all players simultaneously announce a positive integer of their choosing. The smallest unique positive integer wins. What's the optimal strategy for this game? What strategy is so good that, even if everybody was using it - and everybody knew everybody was using it - nobody would change? Such a strategy gives us what's called a Nash equilibrium, and we'll find several for this game. #gametheory #mathematics #mathpuzzle extended cut with algebra: https://youtu.be/7piAPCF5J4g Original Tweet: https://x.com/alz_zyd_/status/1881158525134549024 Join Wrath of Math to get exclusive videos, lecture notes, and more: https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join More math chats: https://www.youtube.com/playlist?list=PLztBpqftvzxXQDmPmSOwXSU9vOHgty1RO ★DONATE★ ◆ Support Wrath of Math on Patreon: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● TikTok: https://www.tiktok.com/@wrathofmathedu ● X: https://x.com/wrathofmathedu 0:00 Intro 2:31 Nash Equilibrium 6:06 Mixed Strategies 8:04 Finding a Trivial Equilibrium 9:41 Symmetric Nash Equilibrium 15:35 The Solution (Part 1) 22:16 The Solution (Part 2) 26:35 Algebra 27:52 Final Details 28:45 Conclusion