War

This game is based on the Blotto game in game theory(you may have come across a version of it if you've seen FiveThirtyEight's "Battle for Riddler Nation" events), with an extra mechanic surrounding raising troops added in order to make the game fun to play over multiple rounds.

• This game has 6 players, and will have 7 rounds.
• Each player starts with 300 regular drones, and 50 special drones. Regular drones can be used every round, while special drones can only be used once each in the game(think of them as self-destructing at the end of the round they are deployed).
• There is one central island(labelled 0), and 6 surrounding islands(labelled 1 through 6), and the distant island(labelled 7).
• For each 2 drones a player deploys to island 0 in a round, they are awarded an extra drone they can use in future rounds.
• For each of islands 1-6, in each round, players are ranked according to the number of drones they send to the island. A player who is ranked Nth on an island X is awarded X*(6-N) points.
• Island 7 is special - it is a colosseum, where players' drones fight to the death. All drones that are sent to island 7 each round will die(which is not the case for the other islands), and be removed from the game. Island 7 awards 50 points to the winner, and 0 points to everybody else.
• There is a final face-off between the combined remaining armies of the players and the dealer. The dealer's army size is fixed at 13000 drones.
• If the dealer's army is larger than the sum total of all the players' armies, they are the sole winner. Otherwise, the player(s) with the most points is(are) the sole winner(s).
• If players fail to send me an action for a round, their actions will default to all their drones doing nothing for the round.
• The individual scores and total drone count at the end of each round are published publicly.
• Private communication is allowed. Feel free to conspire to your heart's content! However, in online versions, sharing screenshots of communications with the dealer is not allowed.

When I ran this game with friends, the players cleared the 13000 drone check, meaning one of them won. There was a fair amount of jostling on the leaderboard, keeping things interesting.

Fun games to try: The Contraband Game, Accelerated Edition

Inspired by the game of the same name from Liar Game. The changes made to the rules allow the game to be played much faster, and the addition of an element from Ultimate Survivor Kaiji's RPS gamble will affect player strategies.

• There are 6 players, and 6 rounds to the game.
• Players are divided into 2 teams of 3 at the start. Team 1 is The Northern Country, team 2 is The Southern Country.
• In each round, each team will be allowed to take 3 briefcases through customs, with the role of customs officer played by a player from the opposing country(players can be the carrier/officer multiple times within a single round). The carrier of each briefcase and the customs officer is determined in the previous round through a second-price auction within their team - i.e. each player makes a sealed bid for any of the briefcases/customs officer roles they are interested in, and the highest bid gets the job for that round for that slot, and that player has to pay the second highest bid for that slot(defaulting to 0 if nobody else bid on it). If nobody bids for a briefcase, it will be considered a no-show, and the associating smuggling chance will not happen.
• Each briefcase can hold 0-100 credits.
• The identity of the smugglers and officers will be announced before players send in their smuggling choices/guesses for that round.
• In all, a total of 6 smuggling attempts happen each round(3 from each team), barring no-shows.
• The customs officer gets to make one guess as to the number of credits in the briefcase that is being carried past them.
• If they guess 0, any money in the briefcase is credited into the associated accounts in Central Country.
• To guess 0 < X <= 100, they must have at least X/2 credits available in Central Country. If the briefcase is empty, or if the briefcase contains more than X credits, the customs officer must pay the smuggler X/2 credits in Central Country as an indemnity. If the briefcase contains non-zero <= X credits, the smuggler must surrender all the money in the briefcase to the customs officer, and it is directly credited to the customs officer's account in Central.
• At the start of the game, each player has 300 credits in their ATM in their own country, and has 100 credits in their Central Country account. The 100 credits is from a loan, which compounds at the end of every round at a rate of 26%.
• At any time, players may withdraw or deposit credits they have in hand from/to their own ATM. There is no limit on how many credits they can have in hand at any given time.
• Players can freely give credits they have in hand to each other. However, credits received in this manner will not be tied to you until they are first deposited into your ATM(after which they can be freely withdrawn).
• When played online, to send credits to another player, both sender and receiver must send the dealer a message with the amount being transferred, the sender's name and the recipient's name. If the sender is holding undeposited money they received from another player, they must specify which money they wish to send(the receiver will not be informed of the person the money is keyed to, and will only be told it's a separate money set).
• At the end of each round, players may spend their credits in Central on paying off as much of their loan as they wish to. This is not publicly announced.
• The team that has made the most money from smuggling + customs is considered the "winner". The smuggling + customs score will be published every round to help players keep track.
• At the end of the game, any money that has not been smuggled will be confiscated, and equally distributed among the players of the opposite team of the one who the money is keyed to. This money does not count towards the score.
• Any assets in Central at the end of the game will first be diverted towards paying off the loan. If this results in a player still being in debt at the end of the game, they automatically lose.
• After all debts are paid off, the members of the "winning" team who end in the black must pay half of their profits back to the dealer.
• At the end of the game, the players are ranked in order of their profits.
• Private communication is freely allowed, so conspire to your heart's content! However, when playing online, sharing screenshots of communications with the dealer is strictly forbidden, and will result in all your money being confiscated if it comes to light.

The rule about interest seems harsh, but if you do the math, you'll realize that assuming no early payments, at the end of the game, the amount to be returned is the same as in Liar Game(ok you have to pay 0.15 more). The interest rule allows players to score more! It also incentivizes players to not stall, to keep things moving along.

I ran the game with friends, and the result was that one team scored far more money, but much like in Liar Game, the winner was someone whose team "lost", but who scored the most money after factoring in the rule about returning half the winnings.

Fun games to try: The Greater Good Game, 7 player edition

Inspired by the game of the same name from Kakegurui. The changes made to the rules make it only really viable to be played once, but it can be a fun experience with friends to see how long it takes for the players to solve the game.

• There are 7 players, and 7 rounds to the game.
• Every round, every player is given 7 (indivisible) coins in hand. They can allot these coins as they wish amongst the following options:
• The public tax box: After each round, the coins in the tax box are counted, and the number is publicly revealed. The coins are then doubled, and distributed equally to all players who are in the game(rounded down) as that many points.
• The private bank: Money deposited in a private bank is not publicly revealed, and is credited directly to your score.
• The casino: If multiple people choose to put money in the casino in a single round, nobody gets any money from the casino. However, if only one person puts money in the casino, they are awarded 3x the money they put in as payout to their score. If the casino awards money to someone, this will be announced in public, but who it is and how much they got will not be.
• The insurance firm: If 0 < N < 4 people put coins in the tax box, the payout is (4-N) times the amount you put into insurance. Else, you get nothing(note that if nobody puts money in the tax box, the payout from insurance is *0*). Insurance payouts are not public.
• 2 times in the game, if 4 players agree, they can vote to exile a player from the game. The exiled player can no longer participate in the game, and their score is set to zero.
• At the end of 7 rounds, if all players have a score < 78, the dealer wins. Otherwise, the highest scoring player wins.
• Private communication between players is *allowed*. Feel free to conspire with each other to your heart's content! However, if playing online, sharing screenshots of communications with the dealer is not allowed.

I've run the game once with friends, and it ended up being a close finish, the "solution" to the game being deduced just a bit too late for the players to beat the dealer.

The "solution" is not quite a Nash equilibrium, but it is quite close, and interesting to try and figure out on your own.

The n-duel problem, solved for n = 4 and generalised

The n-duel problem is something those of us who solve puzzles may have come across in a simplified manner. There are n archers/gunmen/assassins/whatever(I'll go with archers for the purpose of this post) who agree to participate in a turn-based battle. Each of them have 1/n, 2/n, 3/n...n/n probability of killing their target when they shoot. The rules of their engagement are:
1.If there is one person left alive, that person is the winner.
2.If there are multiple people left alive, at any time, it is the turn of the person alive who has shot the least number of arrows to shoot.
3.If there are multiple people who satisfy this criterion, it is the turn of the person who has the lesser accuracy to shoot.

The problem is to determine the ideal strategy for the a/g/a with accuracy 1/n, given that all archers are perfectly rational.

If you haven't solved the puzzle for n=3, stop now and try it. This post will include the answer to it.

I originally saw the generalised version of the puzzle on puzzling.stackexchange.

Hello World!

Hello world! I'm a random person from somewhere in the middle of somewhere, and I have opinions.

That's right, opinions *gasp*. That I don't necessarily always want to keep to myself. Since blogs are things in this day and age, I thought I could occasionally put things up for the rest of the world to see.

I won't promise any kind of schedule, because I know I won't keep it, but I'll try to put things up as often as I can.

To start this off, I'll put up a post about the n-way archer problem.